Maximum Genus, Degree of Vertex and Girth
最大亏格、点度和围长

Let G be a graph. Denote by g(G) the girth of G, and by \delta(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\in C satisfying the condition:d_G(x)>\frac{|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, then \xi(G)\le \max\{1,m\}, k=1, \max\{1,\frac{1}{k-1}m-1\},k=2,3, where m=\frac{|V(G)|g(G)-6}{g(G)^{2}+(\delta(G)-2)g(G)-4}.Moreover, the upper bound is best possible, and a better lower bound of the maximum genus is given.