Let G be an edge-colored connected graph. A path P is a proper path in G if no two adjacent edges of P are colored the same. If P is a proper u − v path of length d(u, v), then P is a proper u − v geodesic. An edge coloring c is a proper-path coloring of a connected graph G if every pair u, v of distinct vertices of G are connected by a proper u − v path in G, and c is a strong proper-path coloring if every two vertices u and v are connected by a proper u− v geodesic in G. The minimum number of colors required for a proper-path coloring or strong proper-path coloring of G is called the proper connection number pc(G) or strong proper connection number spc(G) of G, respectively. If G is a nontrivial connected graph of size m, then pc(G) ≤ spc(G) ≤ m and pc(G) = m or spc(G) = m if and only if G is the star of size m. In this paper, we determine all connected graphs G of size m for which pc(G) or spc(G) is m − 1,m − 2 or m − 3.
@article{bwmeta1.element.doi-10_7151_dmgt_1867,
author = {Chira Lumduanhom and Elliot Laforge and Ping Zhang},
title = {Characterizations of Graphs Having Large Proper Connection Numbers},
journal = {Discussiones Mathematicae Graph Theory},
volume = {36},
year = {2016},
pages = {439-453},
zbl = {1338.05088},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_7151_dmgt_1867}
}
Chira Lumduanhom; Elliot Laforge; Ping Zhang. Characterizations of Graphs Having Large Proper Connection Numbers. Discussiones Mathematicae Graph Theory, Tome 36 (2016) pp. 439-453. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_7151_dmgt_1867/