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- W1629130479 abstract "A path P in an edge-colored graph G is called a proper path if no two adjacent edges of P are colored the same, and G is proper connected if every two vertices of G are connected by a proper path in G. The proper connection number of a connected graph G, denoted by $$textit{pc}(G)$$ , is the minimum number of colors that are needed to make G proper connected. In this paper, we investigate the proper connection number of the complement of a graph G according to some constraints of G itself. Also, we characterize the graphs on n vertices that have proper connection number $$n-2$$ . Using this result, we give a Nordhaus–Gaddum-type theorem for the proper connection number. We prove that if G and $$overline{G}$$ are both connected, then $$4le textit{pc}(G)+textit{pc}(overline{G})le n$$ , and the upper bound holds if and only if G or $$overline{G}$$ is the n-vertex tree with maximum degree $$n-2$$ ." @default.
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- W1629130479 date "2016-05-14" @default.
- W1629130479 modified "2023-09-27" @default.
- W1629130479 title "Proper Connection Numbers of Complementary Graphs" @default.
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- W1629130479 doi "https://doi.org/10.1007/s40840-016-0381-8" @default.
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