On Two Conjectures About the Proper Connection Number of Graphs

Document Type

Article

Publication Date

9-2017

Publication Title

Discrete Mathematics

DOI

10.1016/j.disc.2017.04.022

ISSN

0012-365X

Abstract

A path in an edge-colored graph is called proper if no two consecutive edges of the path receive the same color. For a connected graph G, the proper connection number pc(G) of G is defined as the minimum number of colors needed to color its edges so that every pair of distinct vertices of G is connected by at least one proper path in G. In this paper, we consider two conjectures on the proper connection number of graphs. The first conjecture states that if G is a noncomplete graph with connectivity κ(G) = 2 and minimum degree δ(G) ≥ 3, then pc(G) = 2, posed by Borozan et al. (2012). We give a family of counterexamples to disprove this conjecture. However, from a result of Thomassen it follows that 3-edge-connected noncomplete graphs have proper connection number 2. Using this result, we can prove that if G is a 2-connected noncomplete graph with diam(G) = 3, then pc(G) = 2, which solves the second conjecture we want to mention, posed by Li and Magnant (2015).

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