Strongly Proper Connected Coloring of Graphs

Abstract

We study a new variant of connected coloring of graphs based on the concept of strong edge coloring (every color class forms an induced matching). In particular, an edge-colored path is strongly proper if its color sequence does not contain identical terms within a distance of at most two. A strong proper connected coloring of G is the one in which every pair of vertices is joined by at least one strongly proper path. Let \({{\,\textrm{spc}\,}}(G)\) denote the least number of colors needed for such coloring of a graph G. We prove that the upper bound \({{\,\textrm{spc}\,}}(G)\le 5\) holds for any 2-connected graph G. On the other hand, we demonstrate that there are 2-connected graphs with arbitrarily large girth satisfying \({{\,\textrm{spc}\,}}(G)\ge 4\). Additionally, we prove that graphs whose cycle lengths are divisible by 3 satisfy \({{\,\textrm{spc}\,}}(G)\le 3\). We also consider briefly other connected colorings defined by various restrictions on color sequences of connecting paths. For instance, in a nonrepetitive connected coloring of G, every pair of vertices should be joined by a path whose color sequence is nonrepetitive, that is, it does not contain two adjacent identical blocks. We demonstrate that 2-connected graphs are 15-colorable, while 4-connected graphs are 6-colorable, in the connected nonrepetitive sense. A similar conclusion with a finite upper bound on the number of colors holds for a much wider variety of connected colorings corresponding to fairly general properties of sequences. We end the paper with some open problems of concrete and general nature.