1. Proper Connection Numbers of Complementary Graphs.
- Author
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Huang, Fei, Li, Xueliang, and Wang, Shujing
- Subjects
- *
GRAPH coloring , *PATHS & cycles in graph theory , *GRAPH connectivity , *GEOMETRIC vertices , *MATHEMATICAL bounds - 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 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 G¯ are both connected, then 4≤pc(G)+pc(G¯)≤n , and the upper bound holds if and only if G or G¯ is the n-vertex tree with maximum degree n-2 . [ABSTRACT FROM AUTHOR] - Published
- 2018
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