Wenger＇s graph Hm（q） is a q-regular bipartite graph of order 2qm constructed by using the mdimensional vector space Fq^m over the finite field Fq. The existence of the cycles of certain even length plays an importa...Wenger＇s graph Hm（q） is a q-regular bipartite graph of order 2qm constructed by using the mdimensional vector space Fq^m over the finite field Fq. 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; C2m） 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（Fq） （where ch（Fq） is the character of the finite field Fq） and any vertex v in the Wenger＇s graph Hm（q）, there is a cycle of length 21 in Hm（q） passing through the vertex v.展开更多
基金the National Natural Science Foundation of China(No.10331020,10601038)
文摘Wenger＇s graph Hm（q） is a q-regular bipartite graph of order 2qm constructed by using the mdimensional vector space Fq^m over the finite field Fq. 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; C2m） 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（Fq） （where ch（Fq） is the character of the finite field Fq） and any vertex v in the Wenger＇s graph Hm（q）, there is a cycle of length 21 in Hm（q） passing through the vertex v.
