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

  • Yuefang SunEmail author
Part of the Springer Optimization and Its Applications book series (SOIA, volume 139)


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.



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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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsShaoxing UniversityZhejiangPeople’s Republic of China

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