The Existence of Even Cycles with Specific Lengths in Wenger's Graph

Wenger’s graph H(q)is a q-regular bipartite graph of order 2q~m constructed by using the m- dimensional vector space F~m over the finite field F.The existence of the cycles of certain even length plays an important role in the study of the accurate order of the Turan number ex(n;C)in extremal graph theory. In this paper,we use the algebraic methods of linear system of equations over the finite field and the"critical zero-sum sequences"to show that:if m≥3,then for any integer l with l≠5,4≤l≤2ch(F)(where ch(F) is the character of the finite field F)and any vertex v in the Wenger’s graph H(q),there is a cycle of length 2l in H(q)passing through the vertex v.