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- W2963812438 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 $$6le tpc(G)+tpc(overline{G})le n+2$$ . Examples are given to show that the lower bound is sharp for $$nge 4$$ . The upper bound is reached for $$nge 4$$ if and only if G or $$overline{G}$$ is the tree with maximum degree $$n-2$$ ." @default.
- W2963812438 created "2019-07-30" @default.
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- W2963812438 date "2017-06-20" @default.
- W2963812438 modified "2023-10-17" @default.
- W2963812438 title "Nordhaus–Gaddum-Type Theorem for Total-Proper Connection Number of Graphs" @default.
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- W2963812438 doi "https://doi.org/10.1007/s40840-017-0516-6" @default.
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