Chromatic-connection in graphs (Q2799855)

The primary concepts studied in this paper are the proper connection number \(\operatorname{pc}(G)\) and strong proper connection number \(\operatorname{spc}(G)\) of a graph \(G\). These concepts were, according to the authors, ``inspired by rainbow colorings and proper edge colorings in graphs'' and were introduced by the authors in an earlier paper.NEWLINENEWLINE A path \(P\) in an edge-colored is graph is proper if no two adjacent edges of \(P\) are colored the same. An edge coloring of \(G\) is a proper-path coloring if every pair of vertices of \(G\) are connected by a proper path and is a strong proper-path coloring if every pair of \(u\), \(v\), of vertices are connected by a proper path of length \(d(u,v)\).NEWLINENEWLINE In the introductory paper, these connection numbers were determined for cycles (2 or 3, depending on the parity of the order) and trees (the maximum degree). In this paper, the authors study the proper connection numbers and strong proper connection numbers of some unicyclic graphs, line graphs and powers of graphs.NEWLINENEWLINE The authors then give a general framework introduced by Chartrand but provide no results.