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

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## Abstract

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 \(n-2\).

## Keywords

Total-proper path Total-proper connection number Complementary graph Nordhaus–Gaddum-type## Mathematics Subject Classification

05C15 05C35 05C38 05C40## Notes

### Acknowledgements

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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