Heavy subgraphs, stability and hamiltonicity

TL;DR

This paper introduces a new graph property called S-c-heavy and characterizes graphs where o-heavy and S-c-heavy conditions guarantee hamiltonicity, extending and strengthening previous theorems in graph theory.

Contribution

The paper defines the S-c-heavy property and characterizes all graphs S for which o-heavy and S-c-heavy conditions imply hamiltonicity, providing a new proof and improvements of prior results.

Findings

Characterization of all graphs S with hamiltonian conditions

A new proof of Hu's 1999 theorem

Extension of previous hamiltonicity results

Abstract

Let $G$ be a graph. Adopting the terminology of Broersma et al. and \v{C}ada, respectively, we say that $G$ is 2-heavy if every induced claw ($K_{1,3}$) of $G$ contains two end-vertices each one has degree at least $|V(G)|/2$; and $G$ is o-heavy if every induced claw of $G$ contains two end-vertices with degree sum at least $|V(G)|$ in $G$. In this paper, we introduce a new concept, and say that $G$ is \emph{$S$-c-heavy} if for a given graph $S$ and every induced subgraph $G'$ of $G$ isomorphic to $S$ and every maximal clique $C$ of $G'$, every non-trivial component of $G'-C$ contains a vertex of degree at least $|V(G)|/2$ in $G$. In terms of this concept, our original motivation that a theorem of Hu in 1999 can be stated as every 2-connected 2-heavy and $N$-c-heavy graph is hamiltonian, where $N$ is the graph obtained from a triangle by adding three disjoint pendant edges. In this paper, we will characterize all connected graphs $S$ such that every 2-connected o-heavy and $S$-c-heavy graph is hamiltonian. Our work results in a different proof of a stronger version of Hu's theorem. Furthermore, our main result improves or extends several previous results.