关键词: 交叉数, 联图, 完全多部图, 圈图

Abstract: 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)+n2+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+n2+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：[cr?(K1,1,m∨Cn)≥Z(m+2,n)+][m+12m2n2+m2m-12n2][+m+1,m≥4.]

Key words: crossing numbers, join products, complete multipartite graph, cycle