联图K_(1,1,3)∨C_n的交叉数

Crossing numbers of join products of K_(1,1,3)∨C_n

联图G∨H表示将G的每个顶点与H的每个顶点连边得到的图。在Klesc给出的联图K_(1,1,2)∨C_n的交叉数为Z(4,n)+n/2+3的基础上,根据联图的相关性质,运用反证法和排除法,得到了联图K_(1,1,3)∨C_n与{K_(1,1,3)+e}∨C_n的交叉数均为Z(5,n)+n+n/2+4。并假设在Zarankiewicz猜想成立的前提下,提出对K_(1,1,m)∨C_n(m≥4)的交叉数的一个猜想:cr?(K_(1,1,m)∨C_n)≥Z(m+2,n)+(m+1)/m/2 n/2+m/2 (m-1)/2 n/2+m+1,m≥4。

A join graph denoteted by G∨H,is illustrated by connecting each vertex of graph G to each vertex of graph H.Based on the result that the crossing numbers of K1,1,2∨Cn is Z(4,n)+[n/2]+3 obtained by Klesc,obtain that the crossing numbers of join products K1,1,3∨Cn as well as{K1,1,3+e}∨Cn are Z(5,n)+n+[n/2]+4.The proofs depend on the properties about the join products,and using reduction to absurdity and elimination method.Moreover,a conjecture is given on the crossing number of K1,1,m∨Cn(m≥4)within the conjecture of Zarankiewicz:crΦ(K1,1,m∨Cn)≥Z(m+2,n)+[m+1/2][m/2][n/2]+[m/2][m-1/2][n/2]+m+1,m≥4.