On Minimum Spanning Subgraphs of Graphs With Proper Connection Number 2

Authors

Zhenming BiFollow
Gary Chartrand, Western Michigan UniversityFollow
Garry L. JohnsFollow
Ping ZhangFollow

Publication Date

2016

Abstract

An edge coloring of a connected graph G is a proper-path coloring if every two vertices of G are connected by a properly colored path. The minimum number of colors required of a proper-path coloring of G is called the proper connection number pc(G) of G. For a connected graph G with proper connection number 2, the minimum size of a connected spanning subgraph H of G with pc(H) = 2 is denoted by μ(G). It is shown that if s and t are integers such that t ≥ s + 2 ≥ 5, then μ(K{_s,t) = 2t − 2. We also determine μ(G) for several classes of complete multipartite graphs G. In particular, it is shown that if G = Kn₁, n₂, ..., n_k is a complete k-partite graph, where k ≥ 3, r = Σ^k−1_i=1 n_i ≥ 3 and t = n_k ≥ r² + r, then μ(G) = 2t − 2r + 2

Creative Commons License

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

Recommended Citation

Bi, Zhenming; Chartrand, Gary; Johns, Garry L.; and Zhang, Ping (2016) "On Minimum Spanning Subgraphs of Graphs With Proper Connection Number 2," Theory and Applications of Graphs: Vol. 3: Iss. 2, Article 2.
DOI: 10.20429/tag.2017.030202
Available at: https://digitalcommons.georgiasouthern.edu/tag/vol3/iss2/2