Home > Journals > TAG > Vol. 8 (2021) > Iss. 2

#### Article Title

#### Abstract

An edge-colored graph is properly connected if for every pair of vertices u and v there exists a properly colored uv-path (i.e. a uv-path in which no two consecutive edges have the same color). The proper connection number of a connected graph G, denoted pc(G), is the smallest number of colors needed to color the edges of G such that the resulting colored graph is properly connected. An edge-colored graph is flexibly connected if for every pair of vertices u and v there exist two properly colored paths between them, say P and Q, such that the first edges of P and Q have different colors and the last edges of P and Q have different colors. The flexible connection number of a connected graph G, denoted fpc(G), is the smallest number of colors needed to color the edges of G such that the resulting colored graph is flexibly connected. In this paper, we demonstrate several methods for constructing graphs with pc(G) = 2 and fpc(G) = 2. We describe several families of graphs such that pc(G) ≥ 2 and we settle a conjecture from [BFG+12]. We prove that if G is connected and bipartite, then pc(G) = 2 is equivalent to being 2-edge-connected and fpc(G) = 2 is equivalent to the existence of a path through all cut-edges. Finally, it is proved that every connected, k-regular, Class 1 graph has flexible connection number 2.

#### Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 License.

#### Recommended Citation

Faudree, Jill; Berman, Leah; Chappell, Glenn; Hartman, Chris; Gimbel, John; and Williams, Gordon
(2021)
"On Graphs with Proper Connection Number 2,"
*Theory and Applications of Graphs*: Vol. 8:
Iss.
2, Article 2.

DOI: 10.20429/tag.2021.080202

Available at:
https://digitalcommons.georgiasouthern.edu/tag/vol8/iss2/2

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