On Some Cycles in Wenger Graphs

Abstract

Let p be a prime, q be a power of p, and let F q be the field of q elements. For any positive integer n, the Wenger graph W n (q) is defined as follows: it is a bipartite graph with the vertex partitions being two copies of the (n + 1)-dimensional vector space F n+1 q , 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, W n (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, W n (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 ≤ q c , if 1 ≤ n < (1 − c − 7 3 log q 2)k − 1, then W n (q) contains a 2k-cycle. In particular, W n (q) contains cycles of length 2k, where n + 2 ≤ k ≤ q c , provided q is sufficiently large.