The number of removable edges in a 4-connected graph

  • Authors:

  • Jichang Wu;Xueliang Li;Jianji Su

  • Affiliations:

  • School of Mathematics and System Sciences, Shandong University, Number 27, Shanda South Road, Jinan, Shandong 250100, PR China;Center for Combinatorics, Nankai University, Tianjin 300071, PR China;Department of Mathematics and Computer Science, Guangxi Normal University, Guilin, Guangxi 541004, PR China

  • Venue:
  • Journal of Combinatorial Theory Series B
  • Year:
  • 2004

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Abstract

Let G be a 4-connected graph. For an edge e of G, we do the following operations on G: first, delete the edge e from G, resulting the graph G - e; second, for all the vertices x of degree 3 in G - e, delete x from G - e and then completely connect the 3 neighbors of x by a triangle. If multiple edges occur, we use single edges to replace them. The final resultant graph is denoted by G ⊖ e. If G ⊖ e is still 4-connected, then e is called a removable edge of G. In this paper we prove that every 4-connected graph of order at least six (excluding the 2-cyclic graph of order six) has at least (4|G|+16)/7 removable edges. We also give the structural characterization of 4-connected graphs for which the lower bound is sharp.