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Solution to a Conjecture on the Proper Connection Number of Graphs by Huang, Fei; Li, Xueliang; Qin, Zhongmei; Magnant, Colton is a scholarly article available to read on EtoBox.

What is Solution to a Conjecture on the Proper Connection Number of Graphs about?

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 $pc(G)=2$.

Author

Huang, Fei; Li, Xueliang; Qin, Zhongmei; Magnant, Colton

Published

2016

Language

EN