Pairs of heavy subgraphs for Hamiltonicity of 2-connected graphs

TL;DR

This paper characterizes specific pairs of heavy subgraphs that guarantee Hamiltonicity in 2-connected graphs, extending previous results on subgraph conditions for Hamiltonian cycles.

Contribution

It identifies all pairs of connected graphs (excluding P3) such that 2-connected {R,S}-heavy graphs are necessarily Hamiltonian, broadening understanding of subgraph conditions.

Findings

Characterization of all such graph pairs R and S

Extension of previous forbidden subgraph results

Conditions ensuring Hamiltonicity in 2-connected graphs

Abstract

Let $G$ be a graph on $n$ vertices. An induced subgraph $H$ of $G$ is called heavy if there exist two nonadjacent vertices in $H$ with degree sum at least $n$ in $G$. We say that $G$ is $H$-heavy if every induced subgraph of $G$ isomorphic to $H$ is heavy. For a family $\mathcal{H}$ of graphs, $G$ is called $\mathcal{H}$-heavy if $G$ is $H$-heavy for every $H\in\mathcal{H}$. In this paper we characterize all connected graphs $R$ and $S$ other than $P_3$ (the path on three vertices) such that every 2-connected $\{R,S\}$-heavy graph is Hamiltonian. This extends several previous results on forbidden subgraph conditions for Hamiltonian graphs.