Hamiltonicity in 3-connected claw-free graphs

Abstract

Kuipers and Veldman conjectured that any 3-connected claw-free graph with order ν and minimum degree δ (ν + 6)/10 is Hamiltonian for ν sufficiently large. In this paper, we prove that if H is a 3-connected claw-free graph with sufficiently large order ν, and if δ(H ) (ν + 5)/10, then either H is Hamiltonian, or δ(H ) = (ν + 5)/10 and the Ryjácek's closure cl(H ) of H is the line graph of a graph obtained from the Petersen graph P 10 by adding (ν − 15)/10 pendant edges at each vertex of P 10 .

Degree and neighborhood conditions for hamiltonicity of claw-free graphs

“…For 3-connected claw-free graphs H of order n, Zhang [29] proved that if σ 4 (H) ≥ n − 3, then H is Hamiltonian; Wu [27] proved that if σ 3 (H) ≥ n + 1, then H is Hamiltonian connected. Settling a conjecture posed in [13], Lai et al [18] proved the following: Theorem 1.5 (Lai et al [18]). A 3-connected claw-free simple graph H of order n ≥ 196 with δ(H) ≥ n+5 10 is either Hamiltonian or cl(H) ∈ Q 1 P (n, n−15 10 ).…”

Hamiltonicity and Degrees of Adjacent Vertices in Claw‐Free Graphs

“…Theorem implies Corollary that was a conjecture in and proved by Lai, et al. . Corollary (Lai, et al.…”

“…Corollary (Lai, et al. ). If H is a 3‐connected claw‐free simple graph of order n and n is large enough, and if , then either H is Hamiltonian or and .…”