Proper connection number and connected dominating sets

Abstract: The proper connection number $pc(G)$ of a connected graph $G$ 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 path in $G$ such that no two adjacent edges of the path are colored the same, and such a path is called a proper path. In this paper, we show that for every connected graph with diameter 2 and minimum degree at least 2, its proper connection number is 2. Then, we give an upper bound $\frac{3n}{\delta + 1}-1$ for every connected graph of order $n$ and minimum degree $\delta$. We also show that for every connected graph $G$ with minimum degree at least $2$, the proper connection number $pc(G)$ is upper bounded by $pc(G[D])+2$, where $D$ is a connected two-way (two-step) dominating set of $G$. Bounds of the form $pc(G)\leq 4$ or $pc(G)=2$, for many special graph classes follow as easy corollaries from this result, which include connected interval graphs, asteroidal triple-free graphs, circular arc graphs, threshold graphs and chain graphs, all with minimum degree at least $2$. Furthermore, we get the sharp upper bound 3 for the proper connection numbers of interval graphs and circular arc graphs through analyzing their structures.