MAXIMUM GENUS,DEGREE OF VERTEX AND GIRTH
Huang Yuan-qiu · Xitong kexue yu shuxue · 2009
Let G be a graph.Denote by g(G)the girth of G,and byδ(G)the minimum degree of G.The following two results are proved: 1)Let G be a k-edge-connected simple graph,for any cycle C,there exist a vetex x∈C satisfying the condition: d_G(x)(|V(G)|)/((k-1)~2+2)+k-g(G)+2,k=1,2,3, then G is upper embeddable,and the lower bound is best possible. 2)Let G be a k-edge-connected simple graph,thenξ(G)≤(?)where m=(|V(G)|g(G)-6)/(g(G)~2+(δ(G)-2)g(G)-4′) Moreover,the upper bound is best possible,and a better lower bound of the maximum genus is given.