A Note on the Maximum Genus of Graphs with Diameter 4

Abstract

Let G be a simple graph with diameter four,if G does not contain complete subgraph K3 of order three. We prove that the Betti deficient number of G, �(G) ≤ 2. i.e. the maximum genus of G, M(G) ≥ 1 �(G) − 1 in this paper, which is related with In this paper, G is a finite undirected simple connected graph. The maximum genus γM(G) of G is the largest genus of an orientable surface on which G has a 2-cell embedding, and ξ(G) is the Betti deficiency of G. To determine the maximum genus γM(G) of a graph G on orientable surfaces is related with map geometries, i.e., Smarandache 2-manifolds (see (1) for details) with minimum faces. By Xuong's theory on the maximum genus of a connected graph, ξ(G) equal to β(G) − 2γM(G), where β(G) = |E(G)| − |V (G)| +1 is the Betti number of G. For convenience, we use deficiency to replace the words Betti deficiency in this paper. Nebesky(2) showed that if G is a connected graph and A ⊆ E(G), let υ(G, A) = c(G − A) + b(G − A) − |A| − 1, where c(G − A) denotes the number of components in G − A and b(G − A) denotes the number of components in G − A with an odd Betti number, then we have ξ(G) = max{υ(G, A)|A ⊆ E(G)}. Clearly, the maximum genus of a graph can be determined by its deficiency. In case of that ξ(G) ≤ 1, the graph G is said to be upper embeddable. As we known, following theorems are the main results on relations of the maximum genus with diameter of a graph.