Hamiltonicity in Cherry-quasirandom 3-graphs

The study of Hamilton cycles is a central topic in graph theory with a long and profound history. In recent years, researchers have worked on extending the classical theorem of Dirac on Hamilton cycles to hypergraphs and we refer to [BHS, GPW, HZ1, HZ2, RR, BMSSS1, BMSSS2, RRRSS, HZ_forbidHC, HHZ_cycle] for some recent results and to [KuOs14ICM, RR, zsurvey] for excellent surveys on this topic.

In this paper we restrict ourselves to 3-uniform hypergraphs (3-graphs), where each (hyper)edge contains exactly three vertices. For a 3-graph $H$ and a vertex set $S\subseteq V(H)$, $\deg_{H}(S)$ is defined to be the number of edges containing $S$. The minimum codegree $\delta_{2}(H)$ of $H$ is the minimum of $\deg_{H}(S)$ over all pairs $S$ of vertices in $H$, and the minimum degree $\delta_{1}(H)$ of $H$ is the minimum of $\deg_{H}(v)$ over all vertices $v\in V(H)$. A $3$-graph $C$ is called a tight cycle if its vertices can be ordered cyclically such that every 3 consecutive vertices in this ordering define an edge of $C$, which implies that every two consecutive edges intersect in two vertices. We say that a $3$-graph contains a tight Hamilton cycle if it contains a tight cycle as a spanning subgraph. A tight path $P$ has a sequential order of vertices $v_{1}v_{2}\dots v_{p-1}v_{p}$ such that every 3 consecutive vertices form an edge, where the ends of $P$ are ordered pairs $(v_{2},v_{1})$ and $(v_{p-1},v_{p})$.

The study of quasirandom graphs has been a fruitful area since introduced in [Chung1989, Thomason1987a, Thomason1987b], and we recommend the readers to the excellent survey [Krivelevich2006]. However, the canonical definitions for quasirandom hypergraphs (extending [Chung1989]) have been completely settled only recently [Horev2018, Towsner2017]. In this note we focus on the so-called ‘cherry-quasirandom 3-graphs’ defined as follows. An $n$-vertex 3-graph $H$ is called $(\rho,d)_{\leavevmode\hbox to8.38pt{\vbox to6.53pt{\pgfpicture\makeatletter\hbox{\hskip 1.4143pt\lower-8.8123pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\pgfsys@moveto{1.4143pt}{-7.398pt}\pgfsys@curveto{1.4143pt}{-6.6169pt}{0.7811pt}{-5.98369pt}{0.0pt}{-5.98369pt}\pgfsys@curveto{-0.7811pt}{-5.98369pt}{-1.4143pt}{-6.6169pt}{-1.4143pt}{-7.398pt}\pgfsys@curveto{-1.4143pt}{-8.1791pt}{-0.7811pt}{-8.8123pt}{0.0pt}{-8.8123pt}\pgfsys@curveto{0.7811pt}{-8.8123pt}{1.4143pt}{-8.1791pt}{1.4143pt}{-7.398pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{-7.398pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{-7.398pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\pgfsys@moveto{6.9628pt}{-7.398pt}\pgfsys@curveto{6.9628pt}{-6.6169pt}{6.32959pt}{-5.98369pt}{5.5485pt}{-5.98369pt}\pgfsys@curveto{4.7674pt}{-5.98369pt}{4.13419pt}{-6.6169pt}{4.13419pt}{-7.398pt}\pgfsys@curveto{4.13419pt}{-8.1791pt}{4.7674pt}{-8.8123pt}{5.5485pt}{-8.8123pt}\pgfsys@curveto{6.32959pt}{-8.8123pt}{6.9628pt}{-8.1791pt}{6.9628pt}{-7.398pt}\pgfsys@closepath\pgfsys@moveto{5.5485pt}{-7.398pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.5485pt}{-7.398pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\pgfsys@moveto{4.18855pt}{-3.69899pt}\pgfsys@curveto{4.18855pt}{-2.9179pt}{3.55534pt}{-2.28468pt}{2.77425pt}{-2.28468pt}\pgfsys@curveto{1.99315pt}{-2.28468pt}{1.35994pt}{-2.9179pt}{1.35994pt}{-3.69899pt}\pgfsys@curveto{1.35994pt}{-4.48009pt}{1.99315pt}{-5.1133pt}{2.77425pt}{-5.1133pt}\pgfsys@curveto{3.55534pt}{-5.1133pt}{4.18855pt}{-4.48009pt}{4.18855pt}{-3.69899pt}\pgfsys@closepath\pgfsys@moveto{2.77425pt}{-3.69899pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{2.77425pt}{-3.69899pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{}{{}} {{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{0.96869pt}{-6.10684pt}\pgfsys@lineto{1.80577pt}{-4.99063pt}\pgfsys@stroke\pgfsys@invoke{ } {{}}{}{{}} {{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{3.74297pt}{-4.99062pt}\pgfsys@lineto{4.58008pt}{-6.10677pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$-dense if

for every $\vec{G}_{1},\vec{G}_{2}\subseteq V(H)\times V(H)$, where

Aigner-Horev and Levy proved the following result on tight Hamiltonicity in $(\rho,d)_{\leavevmode\hbox to8.38pt{\vbox to6.53pt{\pgfpicture\makeatletter\hbox{\hskip 1.4143pt\lower-8.8123pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\pgfsys@moveto{1.4143pt}{-7.398pt}\pgfsys@curveto{1.4143pt}{-6.6169pt}{0.7811pt}{-5.98369pt}{0.0pt}{-5.98369pt}\pgfsys@curveto{-0.7811pt}{-5.98369pt}{-1.4143pt}{-6.6169pt}{-1.4143pt}{-7.398pt}\pgfsys@curveto{-1.4143pt}{-8.1791pt}{-0.7811pt}{-8.8123pt}{0.0pt}{-8.8123pt}\pgfsys@curveto{0.7811pt}{-8.8123pt}{1.4143pt}{-8.1791pt}{1.4143pt}{-7.398pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{-7.398pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{-7.398pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\pgfsys@moveto{6.9628pt}{-7.398pt}\pgfsys@curveto{6.9628pt}{-6.6169pt}{6.32959pt}{-5.98369pt}{5.5485pt}{-5.98369pt}\pgfsys@curveto{4.7674pt}{-5.98369pt}{4.13419pt}{-6.6169pt}{4.13419pt}{-7.398pt}\pgfsys@curveto{4.13419pt}{-8.1791pt}{4.7674pt}{-8.8123pt}{5.5485pt}{-8.8123pt}\pgfsys@curveto{6.32959pt}{-8.8123pt}{6.9628pt}{-8.1791pt}{6.9628pt}{-7.398pt}\pgfsys@closepath\pgfsys@moveto{5.5485pt}{-7.398pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.5485pt}{-7.398pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\pgfsys@moveto{4.18855pt}{-3.69899pt}\pgfsys@curveto{4.18855pt}{-2.9179pt}{3.55534pt}{-2.28468pt}{2.77425pt}{-2.28468pt}\pgfsys@curveto{1.99315pt}{-2.28468pt}{1.35994pt}{-2.9179pt}{1.35994pt}{-3.69899pt}\pgfsys@curveto{1.35994pt}{-4.48009pt}{1.99315pt}{-5.1133pt}{2.77425pt}{-5.1133pt}\pgfsys@curveto{3.55534pt}{-5.1133pt}{4.18855pt}{-4.48009pt}{4.18855pt}{-3.69899pt}\pgfsys@closepath\pgfsys@moveto{2.77425pt}{-3.69899pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{2.77425pt}{-3.69899pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{}{{}} {{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{0.96869pt}{-6.10684pt}\pgfsys@lineto{1.80577pt}{-4.99063pt}\pgfsys@stroke\pgfsys@invoke{ } {{}}{}{{}} {{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{3.74297pt}{-4.99062pt}\pgfsys@lineto{4.58008pt}{-6.10677pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$-dense 3-graphs.

They also showed that for $\alpha+d>1$ the $(\rho,d)_{\leavevmode\hbox to8.38pt{\vbox to6.53pt{\pgfpicture\makeatletter\hbox{\hskip 1.4143pt\lower-8.8123pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\pgfsys@moveto{1.4143pt}{-7.398pt}\pgfsys@curveto{1.4143pt}{-6.6169pt}{0.7811pt}{-5.98369pt}{0.0pt}{-5.98369pt}\pgfsys@curveto{-0.7811pt}{-5.98369pt}{-1.4143pt}{-6.6169pt}{-1.4143pt}{-7.398pt}\pgfsys@curveto{-1.4143pt}{-8.1791pt}{-0.7811pt}{-8.8123pt}{0.0pt}{-8.8123pt}\pgfsys@curveto{0.7811pt}{-8.8123pt}{1.4143pt}{-8.1791pt}{1.4143pt}{-7.398pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{-7.398pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{-7.398pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\pgfsys@moveto{6.9628pt}{-7.398pt}\pgfsys@curveto{6.9628pt}{-6.6169pt}{6.32959pt}{-5.98369pt}{5.5485pt}{-5.98369pt}\pgfsys@curveto{4.7674pt}{-5.98369pt}{4.13419pt}{-6.6169pt}{4.13419pt}{-7.398pt}\pgfsys@curveto{4.13419pt}{-8.1791pt}{4.7674pt}{-8.8123pt}{5.5485pt}{-8.8123pt}\pgfsys@curveto{6.32959pt}{-8.8123pt}{6.9628pt}{-8.1791pt}{6.9628pt}{-7.398pt}\pgfsys@closepath\pgfsys@moveto{5.5485pt}{-7.398pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.5485pt}{-7.398pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\pgfsys@moveto{4.18855pt}{-3.69899pt}\pgfsys@curveto{4.18855pt}{-2.9179pt}{3.55534pt}{-2.28468pt}{2.77425pt}{-2.28468pt}\pgfsys@curveto{1.99315pt}{-2.28468pt}{1.35994pt}{-2.9179pt}{1.35994pt}{-3.69899pt}\pgfsys@curveto{1.35994pt}{-4.48009pt}{1.99315pt}{-5.1133pt}{2.77425pt}{-5.1133pt}\pgfsys@curveto{3.55534pt}{-5.1133pt}{4.18855pt}{-4.48009pt}{4.18855pt}{-3.69899pt}\pgfsys@closepath\pgfsys@moveto{2.77425pt}{-3.69899pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{2.77425pt}{-3.69899pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{}{{}} {{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{0.96869pt}{-6.10684pt}\pgfsys@lineto{1.80577pt}{-4.99063pt}\pgfsys@stroke\pgfsys@invoke{ } {{}}{}{{}} {{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{3.74297pt}{-4.99062pt}\pgfsys@lineto{4.58008pt}{-6.10677pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$-denseness together with $\delta_{1}(H)\geq\alpha\binom{n}{2}$ implies tight Hamiltonicity and asked [Horev, Conjecture 1.6] if the condition $\alpha+d>1$ can be dropped. In this note we verify this conjecture.

There are weaker versions of quasirandomness for 3-graphs compared with -denseness, namely, -denseness and -denseness. An $n$-vertex 3-graph $H$ is called $(\rho,d)_{\leavevmode\hbox to8.38pt{\vbox to6.53pt{\pgfpicture\makeatletter\hbox{\hskip 1.4143pt\lower-8.8123pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\pgfsys@moveto{1.4143pt}{-7.398pt}\pgfsys@curveto{1.4143pt}{-6.6169pt}{0.7811pt}{-5.98369pt}{0.0pt}{-5.98369pt}\pgfsys@curveto{-0.7811pt}{-5.98369pt}{-1.4143pt}{-6.6169pt}{-1.4143pt}{-7.398pt}\pgfsys@curveto{-1.4143pt}{-8.1791pt}{-0.7811pt}{-8.8123pt}{0.0pt}{-8.8123pt}\pgfsys@curveto{0.7811pt}{-8.8123pt}{1.4143pt}{-8.1791pt}{1.4143pt}{-7.398pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{-7.398pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{-7.398pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\pgfsys@moveto{6.9628pt}{-7.398pt}\pgfsys@curveto{6.9628pt}{-6.6169pt}{6.32959pt}{-5.98369pt}{5.5485pt}{-5.98369pt}\pgfsys@curveto{4.7674pt}{-5.98369pt}{4.13419pt}{-6.6169pt}{4.13419pt}{-7.398pt}\pgfsys@curveto{4.13419pt}{-8.1791pt}{4.7674pt}{-8.8123pt}{5.5485pt}{-8.8123pt}\pgfsys@curveto{6.32959pt}{-8.8123pt}{6.9628pt}{-8.1791pt}{6.9628pt}{-7.398pt}\pgfsys@closepath\pgfsys@moveto{5.5485pt}{-7.398pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.5485pt}{-7.398pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\pgfsys@moveto{4.18855pt}{-3.69899pt}\pgfsys@curveto{4.18855pt}{-2.9179pt}{3.55534pt}{-2.28468pt}{2.77425pt}{-2.28468pt}\pgfsys@curveto{1.99315pt}{-2.28468pt}{1.35994pt}{-2.9179pt}{1.35994pt}{-3.69899pt}\pgfsys@curveto{1.35994pt}{-4.48009pt}{1.99315pt}{-5.1133pt}{2.77425pt}{-5.1133pt}\pgfsys@curveto{3.55534pt}{-5.1133pt}{4.18855pt}{-4.48009pt}{4.18855pt}{-3.69899pt}\pgfsys@closepath\pgfsys@moveto{2.77425pt}{-3.69899pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{2.77425pt}{-3.69899pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$-dense if

for every $X,Y,Z\subseteq V(H)$; an $n$-vertex 3-graph $H$ is called $(\rho,d)_{\leavevmode\hbox to8.8pt{\vbox to6.81pt{\pgfpicture\makeatletter\hbox{\hskip 1.4143pt\lower-9.38104pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\pgfsys@moveto{1.4143pt}{-7.96674pt}\pgfsys@curveto{1.4143pt}{-7.18564pt}{0.7811pt}{-6.55243pt}{0.0pt}{-6.55243pt}\pgfsys@curveto{-0.7811pt}{-6.55243pt}{-1.4143pt}{-7.18564pt}{-1.4143pt}{-7.96674pt}\pgfsys@curveto{-1.4143pt}{-8.74783pt}{-0.7811pt}{-9.38104pt}{0.0pt}{-9.38104pt}\pgfsys@curveto{0.7811pt}{-9.38104pt}{1.4143pt}{-8.74783pt}{1.4143pt}{-7.96674pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{-7.96674pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{-7.96674pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\pgfsys@moveto{7.38936pt}{-7.96674pt}\pgfsys@curveto{7.38936pt}{-7.18564pt}{6.75615pt}{-6.55243pt}{5.97505pt}{-6.55243pt}\pgfsys@curveto{5.19395pt}{-6.55243pt}{4.56075pt}{-7.18564pt}{4.56075pt}{-7.96674pt}\pgfsys@curveto{4.56075pt}{-8.74783pt}{5.19395pt}{-9.38104pt}{5.97505pt}{-9.38104pt}\pgfsys@curveto{6.75615pt}{-9.38104pt}{7.38936pt}{-8.74783pt}{7.38936pt}{-7.96674pt}\pgfsys@closepath\pgfsys@moveto{5.97505pt}{-7.96674pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.97505pt}{-7.96674pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\pgfsys@moveto{4.40182pt}{-3.98337pt}\pgfsys@curveto{4.40182pt}{-3.20227pt}{3.76862pt}{-2.56906pt}{2.98752pt}{-2.56906pt}\pgfsys@curveto{2.20642pt}{-2.56906pt}{1.57321pt}{-3.20227pt}{1.57321pt}{-3.98337pt}\pgfsys@curveto{1.57321pt}{-4.76447pt}{2.20642pt}{-5.39767pt}{2.98752pt}{-5.39767pt}\pgfsys@curveto{3.76862pt}{-5.39767pt}{4.40182pt}{-4.76447pt}{4.40182pt}{-3.98337pt}\pgfsys@closepath\pgfsys@moveto{2.98752pt}{-3.98337pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{2.98752pt}{-3.98337pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{}{{}} {{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{1.61429pt}{-7.96667pt}\pgfsys@lineto{4.36072pt}{-7.96667pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$-dense if

for every $X\subseteq V(H)$ and $G\subseteq V(H)\times V(H)$. It is known that the -denseness in Theorems 1.1 and 1.2 cannot be replaced by either of these two weaker ones – namely, degenerate choices of $\alpha$ and $d$ do not guarantee tight Hamiltonicity under these two notions of quasirandomness. In this sense, the -denseness in these two theorems is best possible. In contrast, for a weaker notion of Hamiltonicity, namely, the loose cycles, Lenz, Mubayi and Mycroft [Lenz2016] proved that degenerate choices of $\alpha$ and $d$ already force loose Hamiltonicity under -denseness. Very recently, Araújo, Piga and Schacht [Araujo2019] annouced that for any $\alpha>0$ and $d>1/4$, having minimum vertex degree $\alpha\binom{n}{2}$ and being $(\rho,d)_{\leavevmode\hbox to8.8pt{\vbox to6.81pt{\pgfpicture\makeatletter\hbox{\hskip 1.4143pt\lower-9.38104pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\pgfsys@moveto{1.4143pt}{-7.96674pt}\pgfsys@curveto{1.4143pt}{-7.18564pt}{0.7811pt}{-6.55243pt}{0.0pt}{-6.55243pt}\pgfsys@curveto{-0.7811pt}{-6.55243pt}{-1.4143pt}{-7.18564pt}{-1.4143pt}{-7.96674pt}\pgfsys@curveto{-1.4143pt}{-8.74783pt}{-0.7811pt}{-9.38104pt}{0.0pt}{-9.38104pt}\pgfsys@curveto{0.7811pt}{-9.38104pt}{1.4143pt}{-8.74783pt}{1.4143pt}{-7.96674pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{-7.96674pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{-7.96674pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\pgfsys@moveto{7.38936pt}{-7.96674pt}\pgfsys@curveto{7.38936pt}{-7.18564pt}{6.75615pt}{-6.55243pt}{5.97505pt}{-6.55243pt}\pgfsys@curveto{5.19395pt}{-6.55243pt}{4.56075pt}{-7.18564pt}{4.56075pt}{-7.96674pt}\pgfsys@curveto{4.56075pt}{-8.74783pt}{5.19395pt}{-9.38104pt}{5.97505pt}{-9.38104pt}\pgfsys@curveto{6.75615pt}{-9.38104pt}{7.38936pt}{-8.74783pt}{7.38936pt}{-7.96674pt}\pgfsys@closepath\pgfsys@moveto{5.97505pt}{-7.96674pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.97505pt}{-7.96674pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\pgfsys@moveto{4.40182pt}{-3.98337pt}\pgfsys@curveto{4.40182pt}{-3.20227pt}{3.76862pt}{-2.56906pt}{2.98752pt}{-2.56906pt}\pgfsys@curveto{2.20642pt}{-2.56906pt}{1.57321pt}{-3.20227pt}{1.57321pt}{-3.98337pt}\pgfsys@curveto{1.57321pt}{-4.76447pt}{2.20642pt}{-5.39767pt}{2.98752pt}{-5.39767pt}\pgfsys@curveto{3.76862pt}{-5.39767pt}{4.40182pt}{-4.76447pt}{4.40182pt}{-3.98337pt}\pgfsys@closepath\pgfsys@moveto{2.98752pt}{-3.98337pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{2.98752pt}{-3.98337pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{}{{}} {{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{1.61429pt}{-7.96667pt}\pgfsys@lineto{4.36072pt}{-7.96667pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$-dense guarantee tight Hamiltonicity.

In this section we prove the lemmas needed for the proof of Theorem 1.2. Note that by definition, if $H$ is an $n$-vertex $(\rho,d)_{\leavevmode\hbox to8.38pt{\vbox to6.53pt{\pgfpicture\makeatletter\hbox{\hskip 1.4143pt\lower-8.8123pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\pgfsys@moveto{1.4143pt}{-7.398pt}\pgfsys@curveto{1.4143pt}{-6.6169pt}{0.7811pt}{-5.98369pt}{0.0pt}{-5.98369pt}\pgfsys@curveto{-0.7811pt}{-5.98369pt}{-1.4143pt}{-6.6169pt}{-1.4143pt}{-7.398pt}\pgfsys@curveto{-1.4143pt}{-8.1791pt}{-0.7811pt}{-8.8123pt}{0.0pt}{-8.8123pt}\pgfsys@curveto{0.7811pt}{-8.8123pt}{1.4143pt}{-8.1791pt}{1.4143pt}{-7.398pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{-7.398pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{-7.398pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\pgfsys@moveto{6.9628pt}{-7.398pt}\pgfsys@curveto{6.9628pt}{-6.6169pt}{6.32959pt}{-5.98369pt}{5.5485pt}{-5.98369pt}\pgfsys@curveto{4.7674pt}{-5.98369pt}{4.13419pt}{-6.6169pt}{4.13419pt}{-7.398pt}\pgfsys@curveto{4.13419pt}{-8.1791pt}{4.7674pt}{-8.8123pt}{5.5485pt}{-8.8123pt}\pgfsys@curveto{6.32959pt}{-8.8123pt}{6.9628pt}{-8.1791pt}{6.9628pt}{-7.398pt}\pgfsys@closepath\pgfsys@moveto{5.5485pt}{-7.398pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.5485pt}{-7.398pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\pgfsys@moveto{4.18855pt}{-3.69899pt}\pgfsys@curveto{4.18855pt}{-2.9179pt}{3.55534pt}{-2.28468pt}{2.77425pt}{-2.28468pt}\pgfsys@curveto{1.99315pt}{-2.28468pt}{1.35994pt}{-2.9179pt}{1.35994pt}{-3.69899pt}\pgfsys@curveto{1.35994pt}{-4.48009pt}{1.99315pt}{-5.1133pt}{2.77425pt}{-5.1133pt}\pgfsys@curveto{3.55534pt}{-5.1133pt}{4.18855pt}{-4.48009pt}{4.18855pt}{-3.69899pt}\pgfsys@closepath\pgfsys@moveto{2.77425pt}{-3.69899pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{2.77425pt}{-3.69899pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{}{{}} {{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{0.96869pt}{-6.10684pt}\pgfsys@lineto{1.80577pt}{-4.99063pt}\pgfsys@stroke\pgfsys@invoke{ } {{}}{}{{}} {{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{3.74297pt}{-4.99062pt}\pgfsys@lineto{4.58008pt}{-6.10677pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$-dense $3$-graph, then any induced subgraph of $H$ on $\alpha n$ vertices is $(\rho/\alpha^{3},d)_{\leavevmode\hbox to8.38pt{\vbox to6.53pt{\pgfpicture\makeatletter\hbox{\hskip 1.4143pt\lower-8.8123pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\pgfsys@moveto{1.4143pt}{-7.398pt}\pgfsys@curveto{1.4143pt}{-6.6169pt}{0.7811pt}{-5.98369pt}{0.0pt}{-5.98369pt}\pgfsys@curveto{-0.7811pt}{-5.98369pt}{-1.4143pt}{-6.6169pt}{-1.4143pt}{-7.398pt}\pgfsys@curveto{-1.4143pt}{-8.1791pt}{-0.7811pt}{-8.8123pt}{0.0pt}{-8.8123pt}\pgfsys@curveto{0.7811pt}{-8.8123pt}{1.4143pt}{-8.1791pt}{1.4143pt}{-7.398pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{-7.398pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{-7.398pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\pgfsys@moveto{6.9628pt}{-7.398pt}\pgfsys@curveto{6.9628pt}{-6.6169pt}{6.32959pt}{-5.98369pt}{5.5485pt}{-5.98369pt}\pgfsys@curveto{4.7674pt}{-5.98369pt}{4.13419pt}{-6.6169pt}{4.13419pt}{-7.398pt}\pgfsys@curveto{4.13419pt}{-8.1791pt}{4.7674pt}{-8.8123pt}{5.5485pt}{-8.8123pt}\pgfsys@curveto{6.32959pt}{-8.8123pt}{6.9628pt}{-8.1791pt}{6.9628pt}{-7.398pt}\pgfsys@closepath\pgfsys@moveto{5.5485pt}{-7.398pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.5485pt}{-7.398pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\pgfsys@moveto{4.18855pt}{-3.69899pt}\pgfsys@curveto{4.18855pt}{-2.9179pt}{3.55534pt}{-2.28468pt}{2.77425pt}{-2.28468pt}\pgfsys@curveto{1.99315pt}{-2.28468pt}{1.35994pt}{-2.9179pt}{1.35994pt}{-3.69899pt}\pgfsys@curveto{1.35994pt}{-4.48009pt}{1.99315pt}{-5.1133pt}{2.77425pt}{-5.1133pt}\pgfsys@curveto{3.55534pt}{-5.1133pt}{4.18855pt}{-4.48009pt}{4.18855pt}{-3.69899pt}\pgfsys@closepath\pgfsys@moveto{2.77425pt}{-3.69899pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{2.77425pt}{-3.69899pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{}{{}} {{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{0.96869pt}{-6.10684pt}\pgfsys@lineto{1.80577pt}{-4.99063pt}\pgfsys@stroke\pgfsys@invoke{ } {{}}{}{{}} {{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{3.74297pt}{-4.99062pt}\pgfsys@lineto{4.58008pt}{-6.10677pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$-dense and we will use this simple fact without further references.

Throughout the rest of the paper, we will refer to tight paths as just paths. Given a 3-graph $H$, let $\partial H:=\{(x,y):\deg_{H}(xy)>0\}$ be the shadow of $H$. For any $v\in A$, define $\deg_{H}(v,A):=\big{|}N_{H}(v)\cap\binom{A}{2}\big{|}$ and for each pair of vertices $x,y\in A$, let $\deg_{H}(xy,A):=|N_{H}(x,y)\cap A|$.

We use the following result proved in [Han2017, Lemma 8.8]. Note that its original version does not include an estimate on the loss of the number of edges which actually follows from its proof.

The following lemma defines a spanning subgraph of an $n$-vertex $3$-graph $H$, which will be crucial in our proof. We note that in this lemma -denseness is sufficient.

Let $H$ be a $3$-graph. For $v\in V(H)$, a quadruple $(x,y,z,w)\in V(H)^{4}$ is said to be a $v$-absorber if $\{x,y,z\},\{y,z,w\},\{v,x,y\},\{v,y,z\},\{v,z,w\}\in E(H)$. We state and use [Horev, Lemma 4.2] (in a weaker form) and refine the absorbers it gives in the next lemma.

We use the following connection lemma from [Horev].

Now we are ready to prove our absorption lemma.

At last, we use a path cover lemma given in [Horev]. In fact the -denseness is sufficient but we state it in terms of -dense to unify the statement of the lemmas.

Here is a brief sketch of the proof. We first choose a random set $A$, which will be used to deal with the leftover vertices from the almost path cover and connect all paths to a tight cycle. Next, we apply Lemma 2.6 to find an absorbing path $P_{0}$ for $A$, and apply Lemma 2.7 to find a constant number of paths that leaves a set $U$ of vertices uncovered. Using vertices in $A$, we put each vertex in $U$ into disjoint 5-vertex paths, which can be done by applying Lemma 2.6 on $H[A\cup U]$. Now we connect all these paths together into a tight cycle $C$, leaving only some vertices in $A$ outside $V(C)$. Finally the uncovered vertices of $A$ will be absorbed by $P_{0}$ (as $P_{0}$ is an interior path of $C$) and we obtain a tight Hamilton cycle of $H$.

Now we start our proof. Given $\alpha,d\in(0,1]$, let $\sigma=\min\{\frac{1}{132},\frac{d}{33}\}$ and $\zeta=\min\{\frac{\alpha\sigma}{72},\frac{d\sigma}{4320}\}$. Apply Lemma 2.4 with $\alpha/18$ in place of $\alpha$, $d/6$ in place of $d$ and obtain $\rho_{1}$. Apply Lemma 2.5 with $d$ and $\beta=d/20$ and obtain $\rho_{2}$. Apply Lemma 2.6 with $\alpha,d$ and obtain $\rho_{3}$. Apply Lemma 2.7 with $d,\zeta$ and obtain $\rho_{4}$ and $l_{0}$. Let $\rho=\min\{\rho_{1}\sigma^{10}/2^{10},\rho_{2}\sigma^{3}/27,\rho_{3},\rho_{4}^{5}/32,d^{5}/(3^{40}2^{5})\}$ and $n_{0}$ be sufficiently large. Let $n\geq n_{0}$ and $H$ be an $n$-vertex $(\rho,d)_{\leavevmode\hbox to8.38pt{\vbox to6.53pt{\pgfpicture\makeatletter\hbox{\hskip 1.4143pt\lower-8.8123pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\pgfsys@moveto{1.4143pt}{-7.398pt}\pgfsys@curveto{1.4143pt}{-6.6169pt}{0.7811pt}{-5.98369pt}{0.0pt}{-5.98369pt}\pgfsys@curveto{-0.7811pt}{-5.98369pt}{-1.4143pt}{-6.6169pt}{-1.4143pt}{-7.398pt}\pgfsys@curveto{-1.4143pt}{-8.1791pt}{-0.7811pt}{-8.8123pt}{0.0pt}{-8.8123pt}\pgfsys@curveto{0.7811pt}{-8.8123pt}{1.4143pt}{-8.1791pt}{1.4143pt}{-7.398pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{-7.398pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{-7.398pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\pgfsys@moveto{6.9628pt}{-7.398pt}\pgfsys@curveto{6.9628pt}{-6.6169pt}{6.32959pt}{-5.98369pt}{5.5485pt}{-5.98369pt}\pgfsys@curveto{4.7674pt}{-5.98369pt}{4.13419pt}{-6.6169pt}{4.13419pt}{-7.398pt}\pgfsys@curveto{4.13419pt}{-8.1791pt}{4.7674pt}{-8.8123pt}{5.5485pt}{-8.8123pt}\pgfsys@curveto{6.32959pt}{-8.8123pt}{6.9628pt}{-8.1791pt}{6.9628pt}{-7.398pt}\pgfsys@closepath\pgfsys@moveto{5.5485pt}{-7.398pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.5485pt}{-7.398pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\pgfsys@moveto{4.18855pt}{-3.69899pt}\pgfsys@curveto{4.18855pt}{-2.9179pt}{3.55534pt}{-2.28468pt}{2.77425pt}{-2.28468pt}\pgfsys@curveto{1.99315pt}{-2.28468pt}{1.35994pt}{-2.9179pt}{1.35994pt}{-3.69899pt}\pgfsys@curveto{1.35994pt}{-4.48009pt}{1.99315pt}{-5.1133pt}{2.77425pt}{-5.1133pt}\pgfsys@curveto{3.55534pt}{-5.1133pt}{4.18855pt}{-4.48009pt}{4.18855pt}{-3.69899pt}\pgfsys@closepath\pgfsys@moveto{2.77425pt}{-3.69899pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{2.77425pt}{-3.69899pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{}{{}} {{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{0.96869pt}{-6.10684pt}\pgfsys@lineto{1.80577pt}{-4.99063pt}\pgfsys@stroke\pgfsys@invoke{ } {{}}{}{{}} {{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{3.74297pt}{-4.99062pt}\pgfsys@lineto{4.58008pt}{-6.10677pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$-dense $3$-graph. Let $H^{\prime}$ be the spanning subgraph of $H$ given by Lemma 2.2 satisfying (1) and (2).

Define $H^{\prime\prime}:=H^{\prime}[V(H)\setminus(V(P_{0})\cup A)]$. Since $|V(H^{\prime\prime})|\geq n-|V(P_{0})\cup A|\geq n-11|A|\geq 5n/6$, $H^{\prime\prime}$ is $(2\rho^{1/5},d)_{\leavevmode\hbox to8.38pt{\vbox to6.53pt{\pgfpicture\makeatletter\hbox{\hskip 1.4143pt\lower-8.8123pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\pgfsys@moveto{1.4143pt}{-7.398pt}\pgfsys@curveto{1.4143pt}{-6.6169pt}{0.7811pt}{-5.98369pt}{0.0pt}{-5.98369pt}\pgfsys@curveto{-0.7811pt}{-5.98369pt}{-1.4143pt}{-6.6169pt}{-1.4143pt}{-7.398pt}\pgfsys@curveto{-1.4143pt}{-8.1791pt}{-0.7811pt}{-8.8123pt}{0.0pt}{-8.8123pt}\pgfsys@curveto{0.7811pt}{-8.8123pt}{1.4143pt}{-8.1791pt}{1.4143pt}{-7.398pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{-7.398pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{-7.398pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\pgfsys@moveto{6.9628pt}{-7.398pt}\pgfsys@curveto{6.9628pt}{-6.6169pt}{6.32959pt}{-5.98369pt}{5.5485pt}{-5.98369pt}\pgfsys@curveto{4.7674pt}{-5.98369pt}{4.13419pt}{-6.6169pt}{4.13419pt}{-7.398pt}\pgfsys@curveto{4.13419pt}{-8.1791pt}{4.7674pt}{-8.8123pt}{5.5485pt}{-8.8123pt}\pgfsys@curveto{6.32959pt}{-8.8123pt}{6.9628pt}{-8.1791pt}{6.9628pt}{-7.398pt}\pgfsys@closepath\pgfsys@moveto{5.5485pt}{-7.398pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.5485pt}{-7.398pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\pgfsys@moveto{4.18855pt}{-3.69899pt}\pgfsys@curveto{4.18855pt}{-2.9179pt}{3.55534pt}{-2.28468pt}{2.77425pt}{-2.28468pt}\pgfsys@curveto{1.99315pt}{-2.28468pt}{1.35994pt}{-2.9179pt}{1.35994pt}{-3.69899pt}\pgfsys@curveto{1.35994pt}{-4.48009pt}{1.99315pt}{-5.1133pt}{2.77425pt}{-5.1133pt}\pgfsys@curveto{3.55534pt}{-5.1133pt}{4.18855pt}{-4.48009pt}{4.18855pt}{-3.69899pt}\pgfsys@closepath\pgfsys@moveto{2.77425pt}{-3.69899pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{2.77425pt}{-3.69899pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{}{{}} {{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{0.96869pt}{-6.10684pt}\pgfsys@lineto{1.80577pt}{-4.99063pt}\pgfsys@stroke\pgfsys@invoke{ } {{}}{}{{}} {{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{3.74297pt}{-4.99062pt}\pgfsys@lineto{4.58008pt}{-6.10677pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$-dense. By Lemma 2.7, all but at most $\zeta n$ vertices of $H^{\prime\prime}$ can be covered using $l\leq l_{0}$ vertex-disjoint paths $P_{1},P_{2},\dots,P_{l}$. Let $U$ be the set of uncovered vertices of $H^{\prime\prime}$. Let $(a_{i},b_{i})$ and $(c_{i},d_{i})$ be the ends of $P_{i}$ for $i\in[l]$.

We greedily choose disjoint paths $Q_{v}=v_{1}v_{2}vv_{3}v_{4}$ for each $v\in U$ (clearly, a $v$-absorber gives such a path) such that $\deg_{H^{\prime}[A^{*}]}(v_{1}v_{2})\geq d|A^{*}|/18$ and $\deg_{H^{\prime}[A^{*}]}(v_{3}v_{4})\geq d|A^{*}|/18$. Let $W$ be the union of $U\cup\{a_{i},b_{i},c_{i},d_{i}\}_{0\leq i\leq l}$ and the paths that have been chosen so far. Note that $|W|\leq|U|+4(l+1)\leq\zeta n+4l+4\leq\alpha|A^{*}|/72$. Then, for every vertex $v\in U$, we can use the property above to find a $v$-absorber $v_{1}v_{2}v_{3}v_{4}$ in $A^{*}\setminus W=A\setminus W$ such that $\deg_{H^{\prime}[A^{*}]}(v_{1}v_{2})\geq d|A^{*}|/18$ and $\deg_{H^{\prime}[A^{*}]}(v_{3}v_{4})\geq d|A^{*}|/18$.