模n剩余类环的零因子图的补图的类数

Genus of complement of zero-divisor graph for residue class modulo n

研究了模n剩余类环Ζn的零因子图的补图的类数.通过讨论n的素因子个数,利用完全图、完全二部图的类数公式以及有关类数的下界公式和嵌入技巧,证明了模n剩余类环Ζn的零因子图的补图的类数不超过5,当且仅当n=6,8,10,12,14,15,16,18,20,21,22,27,33,35,55,77,p2,其中p为素数.并且分类了模n剩余类环Ζn的零因子图的补图的类数分别为0,1,2,3,4,5的情形.

The genus of complement of zero-divisor graph for residue class modulo n was investigated. According to the prime numbers of n, the genus formulae of complete graph and complete bipartite graph, lower bound of genus graphs and some embedding technique, the genus of complement of zero-divisor graph of residue class modulo n was proved not more than 5 if and only if n equalled to 6,8,10,12,14, 15,16,18,20,21,22,27,33,35,55,77,f. The p meant prime. The classification was completely realized when the genera of complement of zero-divisor graph for residue class modulo n were 0,1,2,3,4,5, respectively.