#### Document Type

Article

#### Publication Date

9-2017

#### Publication Title

Discrete Mathematics

#### First Page

2091

#### Last Page

2107

#### DOI

https://doi.org/10.1016/j.disc.2017.04.010

#### 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 { 9 δ − 3 , n } . In Li (2006), Li conjectured that every 3-connected k -regular claw-free graph H of order n has c ( H ) ≥ min { 10 k − 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 F i ( 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 { 10 δ − 3 , n } or H ∈ F 1 .

(b) either c ( H ) ≥ min { 11 δ − 7 , n } or H ∈ F 1 ∪ F 2 .

(c) either c ( H ) ≥ min { 11 δ − 3 , n } or H ∈ F 1 ∪ F 2 ∪ F 3 .

(d) either c ( H ) ≥ min { 12 δ − 10 , n } or H ∈ F 1 ∪ F 2 ∪ F 3 ∪ F 4 .

(e) if δ ≥ 23 then either c ( H ) ≥ min { 12 δ − 7 , n } or H ∈ F 1 ∪ F 2 ∪ F 3 ∪ F 4 ∪ F 5 .

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).

#### Rights

This is a post-print version of an article originally published in *Discrete Mathematics*, 2017, Volume 340, Issue 9.

The version of record is available through: Elsevier.

#### Recommended Citation

Chen, Zhi-Hong, "Circumferences of 3-connected claw-free graphs, II" *Discrete Mathematics* / (2017): 2091-2107.

Available at https://digitalcommons.butler.edu/facsch_papers/1042