On some cycles in linearized Wenger graphs

Let Fq be the finite field of q elements for prime power q and let p be the character of Fq. For any positive integer m, the linearized Wenger graph Lm(q) is defined as follows: it is a bipartite graph with the vertex partitions being two copies of the (m+1)-dimensional vector space Fqm+1, and two vertices p=(p(1),…,p(m+1)) and l=[l(1),…,l(m+1)] being adjacent if p(i)+l(i)=p(1)(l(1))pi−2, for all i=2,3,…,m+1. In this paper, we show that for any positive integers m and k with 3≤k≤p2, Lm(q) contains even cycles of length 2k which is an open problem put forward by Cao et al. (2015).