Mathematical Sciences: Faculty Publications
On Two Conjectures About the Proper Connection Number of Graphs
Document Type
Article
Publication Date
9-2017
Publication Title
Discrete Mathematics
DOI
10.1016/j.disc.2017.04.022
ISSN
0012-365X
Abstract
A path in an edge-colored graph is called proper if no two consecutive edges of the path receive the same color. For a connected graph G, the proper connection number pc(G) of 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 proper path in G. In this paper, we consider two conjectures on the proper connection number of graphs. The first conjecture states that if G is a noncomplete graph with connectivity κ(G) = 2 and minimum degree δ(G) ≥ 3, then pc(G) = 2, posed by Borozan et al. (2012). We give a family of counterexamples to disprove this conjecture. However, from a result of Thomassen it follows that 3-edge-connected noncomplete graphs have proper connection number 2. Using this result, we can prove that if G is a 2-connected noncomplete graph with diam(G) = 3, then pc(G) = 2, which solves the second conjecture we want to mention, posed by Li and Magnant (2015).
Recommended Citation
Huang, Fei, Xueliang Li, Zhongmei Qin, Colton Magnant, Kenta Ozeki.
2017.
"On Two Conjectures About the Proper Connection Number of Graphs."
Discrete Mathematics, 340 (9): 2217-2222.
doi: 10.1016/j.disc.2017.04.022
https://digitalcommons.georgiasouthern.edu/math-sci-facpubs/591
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