Sharp Nordhaus–Gaddum-Type Lower Bounds for Proper Connection Numbers of Graphs

Yuefang Sun ()
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Yuefang Sun: Shaoxing University

A chapter in Optimization Problems in Graph Theory, 2018, pp 325-331 from Springer

Abstract: Abstract An edge-colored connected graph G is called properly connected if between every pair of distinct vertices, there exists a path that is properly colored. The proper connection number of a connected graph G, denoted by pc(G), is the minimum number of colors needed to color the edges of G to make it properly connected. In this work, we obtain sharp lower bounds for p c ( G ) + p c ( G ¯ ) $$pc(G)+ pc(\overline {G})$$ , and p c ( G ) p c ( G ¯ ) $$pc(G)pc(\overline {G})$$ , where G is a connected graph of order at least 8. Among our results, we also get sharp lower bounds for p v c ( G ) + p v c ( G ¯ ) $$pvc(G)+pvc(\overline {G})$$ , p v c ( G ) p v c ( G ¯ ) $$pvc(G)pvc(\overline {G})$$ , p t c ( G ) + p t c ( G ¯ ) $$ptc(G)+ptc(\overline {G})$$ and p t c ( G ) p t c ( G ¯ ) $$ptc(G)ptc(\overline {G})$$ , where pvc(G) and ptc(G) are proper vertex-connection number and proper total-connection number of G, respectively.

Date: 2018
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DOI: 10.1007/978-3-319-94830-0_13

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