Abstract :
Let T be a spanning tree of a connected graph G. Denote by ξ(G,T) the number of components in G⧹E(T) with odd number of edges. The value minT ξ(G,T) is known as the Betti deficiency of G, denoted by ξ(G), where the minimum is taken over all spanning trees T of G. It is known (N.H. Xuong, J. Combin. Theory 26 (1979) 217–225) that the maximum genus of a graph is mainly determined by its Betti deficiency ξ(G). Let G be a k-edge-connected graph (k⩽3) whose complementary graph has the chromatic number m. In this paper we prove that the Betti deficiency ξ(G) is bounded by a function fk(m) on m, and the bound is the best possible. Thus by Xuongʹs maximum genus theorem we obtain some new results on the lower bounds of the maximum genus of graphs.
