# Proper Vertex Connection and Total Proper Connection

• Xueliang Li
• Colton Magnant
• Zhongmei Qin
Chapter
Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)

## Abstract

Notions of vertex proper connection, the vertex-coloring version of the proper connection number, have been defined and studied independently in Chizmar et al. (AKCE Int J Graphs Comb 13(2):103–106, 2016) and Jiang et al. (Bull Malays Math Sci Soc 41(1):415–425, 2018). A vertex-colored graph G is called proper vertex k-connected if every pair of vertices is connected by k internally disjoint paths, each of which has no two consecutive internal vertices of the same color. Define the proper vertex k-connection number of G, denoted by pvc k (G), to be the smallest number of (vertex) colors needed to make G proper vertex k-connected. We write pvc(G) for pvc1(G). Here the end vertices are not included to be consistent with the similarly defined rainbow vertex connection number where, if end vertices were included, all vertices would necessarily receive distinct colors.

## References

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Chizmar, E., Magnant, C., Salehi Nowbandegani, P.: Note on vertex and total proper connection numbers. AKCE Int. J. Graphs Comb. 13(2), 103–106 (2016)
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Huang, F., Li, X., Wang, S.: Upper bounds of proper connection number of graphs. J. Comb. Optim. 34(1), 165–173 (2017)
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Jiang, H., Li, X., Zhang, Y.: Total proper connection of graphs. arXiv:1512.00726Google Scholar
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Jiang, H., Li, X., Zhang, Y., Zhao, Y.: On (strong) proper vertex-connection of graphs. Bull. Malays. Math. Sci. Soc. 41(1), 415–425 (2018)
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Li, W., Li, X., Zhang, J.: Nordhaus-Gaddum-type theorem for total proper connection number of graphs. Bull. Malays. Math. Sci. Soc. https://doi.org/10.1007/s40840-017-0516-6
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Li, X., Mao, Y., Shi, Y.: The strong rainbow vertex-connection of graphs. Util. Math. 93, 213–223 (2014)

© The Author(s), under exclusive licence to Springer International Publishing AG, part of Springer Nature 2018

## Authors and Affiliations

• Xueliang Li
• 1
• Colton Magnant
• 2
• Zhongmei Qin
• 3
1. 1.Center for CombinatoricsNankai UniversityTianjinChina
2. 2.Department of MathematicsGeorgia Southern UniversityStatesboroUSA
3. 3.College of ScienceChang’an UniversityXi’anChina