The proper connection number of a graph is the least integer k for which the graph has an edge coloring with k colors, with the property that any two vertices are joined by a properly colored path. We prove that given two connected non-bipartite graphs, one of which is (vertex) 2-connected, the proper connection number of their direct product is 2.
@article{bwmeta1.element.doi-10_7151_dmgt_1976,
author = {Richard H. Hammack and Dewey T. Taylor},
title = {Proper Connection Of Direct Products},
journal = {Discussiones Mathematicae Graph Theory},
volume = {37},
year = {2017},
pages = {1005-1013},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_7151_dmgt_1976}
}
Richard H. Hammack; Dewey T. Taylor. Proper Connection Of Direct Products. Discussiones Mathematicae Graph Theory, Tome 37 (2017) pp. 1005-1013. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_7151_dmgt_1976/