Note on minimum degree and proper connection number

Authors

  • Y. Wu Nanjing University
  • Y. Zhang Nanjing University
  • Y. Chen Nanjing University

Keywords:

Proper coloring, Proper connection number, Minimum degree

Abstract

http://dx.doi.org/10.1017/S0004972712000330An edge-colored graph $G$ is called \textit{properly connected} if any two vertices are connected by a properly colored path. The \textit{proper connection number} of a graph $G$, denoted by $pc(G)$, is the smallest number of colors that are needed to color $G$ such that it is properly connected. Let $\delta(n)$ denote the minimum value such that $pc(G)=2$ for any 2-connected incomplete graph $G$ of order $n$ with minimum degree at least $\delta(n)$. Brause et al. showed in [{\it Minimum degree conditions for the proper connection number of graphs, Graphs and Combinatorics, 33(2017), 833-843}] that $\delta(n)>n/42$. In this note, we show that $\delta(n)>n/36$.

Author Biographies

Y. Wu, Nanjing University

Department of Mathematics

Y. Zhang, Nanjing University

Department of Mathematics

Y. Chen, Nanjing University

Department of Mathematics

Published

2021-03-10

Issue

Vol. 103 No. 2 (2021)

Section

Articles
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