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Minimum degree and size conditions for the proper connection number of graphs

Author

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  • Guan, Xiaxia
  • Xue, Lina
  • Cheng, Eddie
  • Yang, Weihua

Abstract

An edge-coloured graph G is called properly connected if every two vertices are connected by a proper path. The proper connection number of a connected graph G, denoted by pc(G), is the smallest number of colours that are needed in order to make G properly connected. van Aardt et al. (2017)gave a sufficient condition for the proper connection number to be at most k in terms of the size of graphs. In this note, our main result is the following, by adding a minimum degree condition: let G be a connected graph of order n, k ≥ 3. If |E(G)|≥(n−m−(k+1−m)(δ+1)2)+(k+1−m)(δ+12)+k+2, then pc(G) ≤ k, where m takes the value k+1 if δ=1 and ⌊kδ−1⌋ if δ ≥ 2. Furthermore, if k=2 and δ=2,pc(G) ≤ 2, except G ∈ {G1, Gn} (n ≥ 8), where G1=K1∨3K2 and Gn is obtained by taking a complete graph Kn−5 and K1∨(2K2) with an arbitrary vertex of Kn−5 and a vertex with d(v)=4 in K1∨(2K2) being joined. If k=2,δ ≥ 3, we conjecture pc(G) ≤ 2, where m takes the value 1 if δ=3 and 0 if δ ≥ 4 in the assumption.

Suggested Citation

  • Guan, Xiaxia & Xue, Lina & Cheng, Eddie & Yang, Weihua, 2019. "Minimum degree and size conditions for the proper connection number of graphs," Applied Mathematics and Computation, Elsevier, vol. 352(C), pages 205-210.
    • Handle: RePEc:eee:apmaco:v:352:y:2019:i:c:p:205-210
    DOI: 10.1016/j.amc.2019.01.062
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    References listed on IDEAS

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    1. Hui Lei & Shasha Li & Henry Liu & Yongtang Shi, 2018. "Rainbow vertex connection of digraphs," Journal of Combinatorial Optimization, Springer, vol. 35(1), pages 86-107, January.
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    Cited by:

    1. Ma, Yingbin & Zhang, Xiaoxue, 2023. "Graphs with (strong) proper connection numbers m−3 and m−4," Applied Mathematics and Computation, Elsevier, vol. 445(C).

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