Induced nets and Hamiltonicity of claw-free graphs

TL;DR

This paper proves Broersma's 1993 conjecture that certain 2-connected claw-free graphs with specific degree conditions on endvertices of induced nets are Hamiltonian, advancing understanding of Hamiltonicity in such graphs.

Contribution

The paper confirms Broersma's conjecture, establishing a new sufficient condition for Hamiltonicity in 2-connected claw-free graphs based on induced nets and degree constraints.

Findings

Broersma's conjecture is proven true.

Degree conditions on endvertices ensure Hamiltonicity.

Advances in understanding Hamiltonian cycles in claw-free graphs.

Abstract

The connected graph of degree sequence 3,3,3,1,1,1 is called a net, and the vertices of degree 1 in a net is called its endvertices. Broersma conjectured in 1993 that a 2-connected graph G with no induced K_{1,3} is hamiltonian if every endvertex of each induced net of G has degree at least (|V(G)|-2)/3. In this paper we prove this conjecture in the affirmative.