Ye WANG, Felix LAZEBNIK, Andrew THOMASON
Let p be a prime, q be a power of p, and let Fq be the field of q elements. For any positive integer n, the Wenger graph Wn(q) is defined as follows:it is a bipartite graph with the vertex partitions being two copies of the (n+1)-dimensional vector space Fqn+1, and two vertices p=(p(1), …, p(n+1)) and l=[l(1), …, l(n+1)] being adjacent if p(i) + l(i)=p(1)l(1)i-1, for all i=2, 3, …, n + 1.
In 2008, Shao, He and Shan showed that for n ≥ 2, Wn(q) contains a cycle of length 2k where 4 ≤ k ≤ 2p and k≠ 5. In this paper we extend their results by showing that
(i) for n ≥ 2 and p ≥ 3, Wn(q) contains cycles of length 2k, where 4 ≤ k ≤ 4p + 1 and k≠ 5;
(ii) for q ≥ 5, 0 < c < 1, and every integer k, 3 ≤ k ≤ qc, if 1 ≤ n < (1-c-7/3 logq 2)k-1, then Wn(q) contains a 2k-cycle. In particular, Wn(q) contains cycles of length 2k, where n + 2 ≤ k ≤ qc, provided q is sufficiently large.
