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Sharp Nordhaus–Gaddum-Type Lower Bounds for Proper Connection Numbers of Graphs

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Optimization Problems in Graph Theory

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 139))

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Abstract

An edge-colored connected graph G is called properly connected if between every pair of distinct vertices, there exists a path that is properly colored. The proper connection number of a connected graph G, denoted by pc(G), is the minimum number of colors needed to color the edges of G to make it properly connected. In this work, we obtain sharp lower bounds for \(pc(G)+ pc(\overline {G})\), and \(pc(G)pc(\overline {G})\), where G is a connected graph of order at least 8. Among our results, we also get sharp lower bounds for \(pvc(G)+pvc(\overline {G})\), \(pvc(G)pvc(\overline {G})\), \(ptc(G)+ptc(\overline {G})\) and \(ptc(G)ptc(\overline {G})\), where pvc(G) and ptc(G) are proper vertex-connection number and proper total-connection number of G, respectively.

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Acknowledgements

This work was supported by National Natural Science Foundation of China (No. 11401389) and China Scholarship Council (No. 201608330111). The author is very grateful to the referee for helpful comments and suggestions.

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Authors and Affiliations

  1. Department of Mathematics, Shaoxing University, Zhejiang, People’s Republic of China

    Yuefang Sun

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  1. Yuefang Sun
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Correspondence to Yuefang Sun .

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Editors and Affiliations

  1. Department of Information Systems and Decision Science, Merrick School of Business, University of Baltimore, Baltimore, MD, USA

    Boris Goldengorin

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Sun, Y. (2018). Sharp Nordhaus–Gaddum-Type Lower Bounds for Proper Connection Numbers of Graphs. In: Goldengorin, B. (eds) Optimization Problems in Graph Theory. Springer Optimization and Its Applications, vol 139. Springer, Cham. https://doi.org/10.1007/978-3-319-94830-0_13

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