Closures and heavy pairs for hamiltonicity

TL;DR

This paper investigates conditions under which certain heavy subgraph structures guarantee Hamiltonicity in graphs, extending previous results through the application of closure concepts and structural characterizations.

Contribution

It generalizes earlier Hamiltonicity conditions by analyzing the structure of $c$-closure in 2-connected $\

Findings

Characterization of the $c$-closure of 2-connected $\

Extension of Hamiltonicity conditions involving forbidden or $o$-heavy subgraphs.

Abstract

We say that a graph $G$ on $n$ vertices is $\{H,F\}$-$o$-heavy if every induced subgraph of $G$ isomorphic to $H$ or $F$ contains two nonadjacent vertices with degree sum at least $n$. Generalizing earlier sufficient forbidden subgraph conditions for hamiltonicity, in 2012, Li, Ryj\'a\v{c}ek, Wang and Zhang determined all connected graphs $R$ and $S$ of order at least 3 other than $P_3$ such that every 2-connected $\{R,S\}$-$o$-heavy graph is hamiltonian. In particular, they showed that, up to symmetry, $R$ must be a claw and $S\in\{P_4,P_5,C_3,Z_1,Z_2,B,N,W\}$. In 2008, \v{C}ada extended Ryj\'a\v{c}ek's closure concept for claw-free graphs by introducing what we call the $c$-closure for claw-$o$-heavy graphs. We apply it here to characterize the structure of the $c$-closure of 2-connected $\{R,S\}$-$o$-heavy graphs, where $R$ and $S$ are as above. Our main results extend or generalize several earlier results on hamiltonicity involving forbidden or $o$-heavy subgraphs.