Edges incident with a vertex of degree greater than four and a lower bound on the number of contractible edges in a 4-connected graph

Abstract

In this paper, we prove that the number of 4-contractible edges (edges that after contraction do not change the connectivity of the initial graph) of a 4-connected graph G is at least $(1/28){\sum }_{x\in V_{\ge 5}(G)}{\mathrm{deg}}_G(x)$, where $V_{\ge 5}(G)$ denotes the set of those vertices of G which have degree greater than or equal to 5.

This is the refinement of the result proved by Ando et al. [On the number of 4-contractible edges in 4-connected graphs, J. Combin. Theory Ser. B 99 (2009) 97–109].