Some extremal results on the colorful monochromatic vertex-connectivity of a graph
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A path in a vertex-colored graph is called a vertex-monochromatic path if its internal vertices have the same color. A vertex-coloring of a graph is a monochromatic vertex-connection coloring (MVC-coloring for short), if there is a vertex-monochromatic path joining any two vertices in the graph. For a connected graph G, the monochromatic vertex-connection number, denoted by mvc(G), is defined to be the maximum number of colors used in an MVC-coloring of G. These concepts of vertex-version are natural generalizations of the colorful monochromatic connectivity of edge-version, introduced by Caro and Yuster (Discrete Math 311:1786–1792, 2011). In this paper, we mainly investigate the Erdős–Gallai-type problems for the monochromatic vertex-connection number mvc(G) and completely determine the exact value. Moreover, the Nordhaus–Gaddum-type inequality for mvc(G) is also given.
KeywordsVertex-monochromatic path MVC-coloring Monochromatic vertex-connection number Erdős–Gallai-type problem Nordhaus–Gaddum-type problem
Mathematics Subject Classification05C15 05C35 05C38 05C40
The authors are very grateful to the reviewers for their useful comments and suggestions, which helped to improve the presentation of the paper.
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