Induced subgraphs with large degrees at end-vertices for hamiltonicity of claw-free graphs

TL;DR

This paper characterizes all graphs H for which a 2-connected claw-free graph G is Hamiltonian if end-vertices of induced copies of H have degrees above a certain threshold, advancing a conjecture in graph theory.

Contribution

It provides a complete characterization of graphs H that ensure Hamiltonicity under degree conditions on end-vertices, confirming Broersma's conjecture up to an additive constant.

Findings

Characterization of all graphs H satisfying the degree condition for Hamiltonicity.

Extension of Broersma's conjecture with an additive constant.

Improved understanding of degree conditions for Hamiltonian cycles in claw-free graphs.

Abstract

A graph is called \emph{claw-free} if it contains no induced subgraph isomorphic to $K_{1,3}$. Matthews and Sumner proved that a 2-connected claw-free graph $G$ is hamiltonian if every vertex of it has degree at least $(|V(G)|-2)/3$. At the workshop C\&C (Novy Smokovec, 1993), Broersma conjectured the degree condition of this result can be restricted only to end-vertices of induced copies of $N$ (the graph obtained from a triangle by adding three disjoint pendant edges). Fujisawa and Yamashita showed that the degree condition of Matthews and Sumner can be restricted only to end-vertices of induced copies of $Z_1$ (the graph obtained from a triangle by adding one pendant edge). Our main result in this paper is a characterization of all graphs $H$ such that a 2-connected claw-free graph $G$ is hamiltonian if each end-vertex of every induced copy of $H$ in $G$ has degree at least $|V(G)|/3+1$. This gives an affirmative solution of the conjecture of Broersma up to an additive constant.