Two Classes of Simple MCD Graphs

Abstract

Let S _n be the set of simple graphs on n vertices in which no two cycles have the same length. A graph G in S _n is called a simple maximum cycle-distributed graph (simple MCD graph) if there exists no graph G′ in S _n with |E(G′)| > |E(G)|. A planar graph G is called a generalized polygon path (GPP) if G ^* formed by the following method is a path: corresponding to each interior face f of $\tilde G$ ($\tilde G$ is a plane graph of G) there is a vertex f ^* of G ^*; two vertices f ^* and g ^* are adjacent in G ^* if and only if the intersection of the boundaries of the corresponding interior faces of $\tilde G$ is a simple path of $\tilde G$. In this paper, we prove that there exists a simple MCD graph on n vertices such that it is a 2-connected graph being not a GPP if and only if n ∈ {10, 11, 14, 15, 16, 21, 22}. We also prove that, by discussing all the natural numbers except for 75 natural numbers, there are exactly 18 natural numbers, for each n of which, there exists a simple MCD graph on n vertices such that it is a 2-connected graph.

This work was supported by the Foundation of the Development of Science and Technology of Shanghai Higher Learning (04DB25) and Natural Science Foundation of China (10571057).