Let p be a prime,q be a power of p,and let Fq be the field of q elements.For any positive integer n,the Wenger graph Wn(q)is defined as follows:it is a bipartite graph with the vertex partitions being two copies of th...Let p be a prime,q be a power of p,and let Fq be the field of q elements.For any positive integer n,the Wenger graph Wn(q)is defined as follows:it is a bipartite graph with the vertex partitions being two copies of the(n+1)-dimensional vector space Fq^n+1,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,Wn(q)contains a cycle of length 2 k where 4≤k≤2 p and k≠5.In this paper we extend their results by showing that(i)for n≥2 and p≥3,Wn(q)contains cycles of length 2k,where 4≤k≤4 p+1 and k≠5;(ii)for q≥5,0<c<1,and every integer k,3≤k≤qc,if 1≤n<(1-c-7/3 logq2)k-1,then Wn(q)contains a 2 k-cycle.In particular,Wn(q)contains cycles of length 2 k,where n+2≤k≤qc,provided q is sufficiently large.展开更多
We introduce a new infinite class of bipartite graphs, called jumped Wenger graphs, which has a similar structure with Wenger graphs. We give a tight upper bound of the diameter for these graphs and the exact diameter...We introduce a new infinite class of bipartite graphs, called jumped Wenger graphs, which has a similar structure with Wenger graphs. We give a tight upper bound of the diameter for these graphs and the exact diameter for some special graphs. We also determine the girth of the jumped Wenger graphs.展开更多
基金supported by NSF grant DMS-1106938-002,NSFC(Nos.11701372.11801371)Shanghai Sailing Program(No.19YF1435500).
文摘Let p be a prime,q be a power of p,and let Fq be the field of q elements.For any positive integer n,the Wenger graph Wn(q)is defined as follows:it is a bipartite graph with the vertex partitions being two copies of the(n+1)-dimensional vector space Fq^n+1,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,Wn(q)contains a cycle of length 2 k where 4≤k≤2 p and k≠5.In this paper we extend their results by showing that(i)for n≥2 and p≥3,Wn(q)contains cycles of length 2k,where 4≤k≤4 p+1 and k≠5;(ii)for q≥5,0<c<1,and every integer k,3≤k≤qc,if 1≤n<(1-c-7/3 logq2)k-1,then Wn(q)contains a 2 k-cycle.In particular,Wn(q)contains cycles of length 2 k,where n+2≤k≤qc,provided q is sufficiently large.
基金The work of L.-P.Wang was supported in part by the National Natural Science Foundation of China(Grant No.61872355)National Key Research and Development Program of China(Grant No.2018YFA0704703).
文摘We introduce a new infinite class of bipartite graphs, called jumped Wenger graphs, which has a similar structure with Wenger graphs. We give a tight upper bound of the diameter for these graphs and the exact diameter for some special graphs. We also determine the girth of the jumped Wenger graphs.
