C_(m,n)的最小亏格与K_(m,n)的强嵌入(英文)

Genus of C_(m,n) and strong genus embedding of K_(m,n)

令C_(m,n)表示长为m的圈与n个孤立点的联结(join)所得的图.本文证明了C_(m,n)的最小亏格和最小不可定向亏格与完全二部图K_(m,n)的相等.同时,证明当m≥2并且n≥2时,K_(m,n)在其最小可定向曲面上有一个强嵌入;当m≥3并且n≥3时,在最小不可定向曲面上有一个强嵌入.

Let C_（m,n） be the join graph of C_m（a cycle of length m） and n isolated vertices. In this paper,we first show that the genus and nonorientable genus of C_（m,n） equal those of K_（m,n）,which were well known and discovered by Ringel[1,2].Then we show that the complete bipartite graph K_（m,n） has a strong orientable genus embedding if m≥2 and n≥2 and has a strong nonorientable genus embedding if m≥3 and n≥3.