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

作　　者：Jia-yu Shao Chang-xiang He Hai-ying Shan 

机构地区：[1]Department of Mathematics, Tongji University, Shanghai 200092, China [2]College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China

出　　处：《Acta Mathematicae Applicatae Sinica》2008年第2期281-288,共8页应用数学学报（英文版）

基　　金：the National Natural Science Foundation of China(No.10331020,10601038)

摘　　要：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.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.

关 键 词：GRAPH CYCLE finite field vector space linear system of equations 

分 类 号：O157.6[理学—数学]