Proper connection number and connected dominating sets

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 3n δ+1 − 1 for every connected graph of order n and minimum degree δ. 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) ≤ 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.