Circumferences of 3-connected claw-free graphs, II

Abstract

For a graph H, the circumference of H, denoted by c(H), is the length of a longest cycle in H. It is proved in Chen (2016) that if H is a 3-connected claw-free graph of order n with δ≥8, then c(H)≥min{9δ−3,n}. In Li (2006), Li conjectured that every 3-connected k-regular claw-free graph H of order n has c(H)≥min{10k−4,n}. Later, Li posed an open problem in Li (2008): how long is the best possible circumference for a 3-connected regular claw-free graph? In this paper, we study the circumference of 3-connected claw-free graphs without the restriction on regularity and provide a solution to the conjecture and the open problem above. We determine five families Fi (1≤i≤5) of 3-connected claw-free graphs which are characterized by graphs contractible to the Petersen graph and show that if H is a 3-connected claw-free graph of order n with δ≥16, then one of the following holds:(a) either c(H)≥min{10δ−3,n} or H∈F1.(b) either c(H)≥min{11δ−7,n} or H∈F1∪F2.(c) either c(H)≥min{11δ−3,n} or H∈F1∪F2∪F3.(d) either c(H)≥min{12δ−10,n} or H∈F1∪F2∪F3∪F4.(e) if δ≥23 then either c(H)≥min{12δ−7,n} or H∈F1∪F2∪F3∪F4∪F5.This is also an improvement of the prior results in Chen (2016), Lai et al. (2016), Li et al. (2009) and Mathews and Sumner (1985).