Nordhaus–Gaddum-Type Theorem for Total-Proper Connection Number of Graphs

  • Wenjing Li
  • Xueliang LiEmail author
  • Jingshu Zhang


A graph is said to be total-colored if all the edges and the vertices of the graph are colored. A path P in a total-colored graph G is called a total-proper path if (1) any two adjacent edges of P are assigned distinct colors; (2) any two adjacent internal vertices of P are assigned distinct colors; and (3) any internal vertex of P is assigned a distinct color from its incident edges of P. The total-colored graph G is total-proper connected if any two distinct vertices of G are connected by a total-proper path. The total-proper connection number of a connected graph G, denoted by tpc(G), is the minimum number of colors that are required to make G total-proper connected. In this paper, we first characterize the graphs G on n vertices with \(tpc(G)=n-1\). Based on this, we obtain a Nordhaus–Gaddum-type result for total-proper connection number. We prove that if G and \(\overline{G}\) are connected complementary graphs on n vertices, then \(6\le tpc(G)+tpc(\overline{G})\le n+2\). Examples are given to show that the lower bound is sharp for \(n\ge 4\). The upper bound is reached for \(n\ge 4\) if and only if G or \(\overline{G}\) is the tree with maximum degree \(n-2\).


Total-proper path Total-proper connection number Complementary graph Nordhaus–Gaddum-type 

Mathematics Subject Classification

05C15 05C35 05C38 05C40 



The authors would like to thank the reviewers for their helpful comments and suggestions, which helped to improve the presentation of the paper.


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Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2017

Authors and Affiliations

  1. 1.Center for Combinatorics and LPMCNankai UniversityTianjinChina
  2. 2.School of Mathematics and StatisticsQinghai Normal UniversityXiningChina

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