最大亏格、点度和围长

用g(G)和δ(G)分别表示一个图G的围长和顶点最小度. ζ(G)为图G的Betii亏数,主要证明了以下2个结果1)设G为k-边连通简单图,若对G中任意圈C,存在点x∈C满足dG(x)＞|V(G)|/(k-1)2+2)+k-g(G)+2,k=1,2,3,则G是上可嵌入的.且不等式的下界是最好的;2)设G为k-边连通简单图,则ζ(G)≤{max{1,m},k=1,max{1,1/(k-1)m -1}K=2,3 其中m= |V(G)|g(G)-6/g(G)2+(δ(G)-2)g(G)-4'且不等式的上界是可达的.进而得到了最大亏格一个比较好的下界.

Maximum Genus, Degree of Vertex and Girth

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.