Two classes of simple MCD graphs

  • Authors:

  • Yong-Bing Shi;Yin-Cai Tang;Hua Tang;Ling-Liu Gong;Li Xu

  • Affiliations:

  • Department of Applied Mathematics, Shanghai Teachers' University, Shanghai, P.R. China;Department of Statistics, East China Normal University, Shanghai, P.R. China;Department of Applied Mathematics, Shanghai Teachers' University, Shanghai, P.R. China;Department of Applied Mathematics, Shanghai Teachers' University, Shanghai, P.R. China;Department of Applied Mathematics, Shanghai Teachers' University, Shanghai, P.R. China

  • Venue:
  • CJCDGCGT'05 Proceedings of the 7th China-Japan conference on Discrete geometry, combinatorics and graph theory
  • Year:
  • 2005

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Abstract

Let Sn be the set of simple graphs on n vertices in which no two cycles have the same length. A graph G in Sn is called a simple maximum cycle-distributed graph (simple MCD graph) if there exists no graph G′ in Sn 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 G (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 G is a simple path of 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.