#### Title

Proper Connection of Graphs

#### Document Type

Article

#### Publication Date

9-6-2012

#### Publication Title

Discrete Mathematics

#### DOI

10.1016/j.disc.2011.09.003

#### ISSN

0012-365X

#### Abstract

An edge-colored graph G is k-proper connected if every pair of vertices is connected by k internally pairwise vertex-disjoint proper colored paths. The k-proper connection number of a connected graph G, denoted by pc_{k}(G), is the smallest number of colors that are needed to color the edges of G in order to make it k-proper connected. In this paper we prove several upper bounds for pc_{k}(G). We state some conjectures for general and bipartite graphs, and we prove them for the case when k=1. In particular, we prove a variety of conditions on G which imply pc_{1}(G)=2.

#### Recommended Citation

Borozan, Valentin, Shinya Fujita, Aydin Gerek, Colton Magnant, Yannis Manoussakis, Leandro Montero, Zsolt Tuza.
2012.
"Proper Connection of Graphs."
*Discrete Mathematics*, 312 (17): 2550-2560.
doi: 10.1016/j.disc.2011.09.003

https://digitalcommons.georgiasouthern.edu/math-sci-facpubs/105