Results of the maximum genus of graphs

In this paper,we provide a new class of up-embeddable graphs,and obtain a tight lower bound on the maximum genus of a class of 2-connected pseudographs of diameter 2 and of a class of diameter 4 multi-graphs.This extends a result of Skoviera.     

Up-embeddability via girth and the degree-sum of adjacent vertices

Let G be a simple graph of order n and girth g. For any two adjacent vertices u and v of G, if d _G (u) + d _G (v) ⩾ n − 2g + 5 then G is up-embeddable. In the case of 2-edge-connected (resp. 3-edge-connected) graph, G is up-embeddable if d _G (u) + d _G (v) ⩾ n − 2g + 3 (resp. d _G (u) + d _G (v) ⩾ n − 2g −5) for any two adjacent vertices u and v of G. Furthermore, the above three lower bounds are all shown to be tight. This work was supported by National Natural Science Foundation of China (Grant No. 10571013)