Solution to a conjecture on the proper connection number of graphs

TL;DR

This paper proves a conjecture that connected noncomplete graphs with sufficiently high minimum degree have a proper connection number of 2, with only two small exceptions, and also applies this to bipartite graphs.

Contribution

It confirms a conjecture on the proper connection number for a broad class of graphs, identifying conditions under which it equals 2, and extends results to bipartite graphs.

Findings

The conjecture holds for all but two small graphs on 7 and 8 vertices.

Connected bipartite graphs with minimum degree at least (n+6)/8 have proper connection number 2.

The paper provides a near-complete characterization of graphs with proper connection number 2 under certain degree conditions.

Abstract

A path in an edge-colored graph is called a proper path if no two adjacent 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 $G$. Recently, Li and Magnant in [Theory Appl. Graphs 0(1)(2015), Art.2] posed the following conjecture: If $G$ is a connected noncomplete graph of order $n \geq 5$ and minimum degree $\delta(G) \geq n/4$, then $pc(G)=2$. In this paper, we show that this conjecture is true except for two small graphs on 7 and 8 vertices, respectively. As a byproduct we obtain that if $G$ is a connected bipartite graph of order $n\geq 4$ with $\delta(G)\geq \frac{n+6}{8}$, then $pc(G)=2$.