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arXiv · 1503.08941

Some extremal results on the colorful monochromatic vertex-connectivity of a graph

Qingqiong Cai·Xueliang Li·Di Wu
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Abstract

A path in a vertex-colored graph is called a \emph{vertex-monochromatic path} if its internal vertices have the same color. A vertex-coloring of a graph is a \emph{monochromatic vertex-connection coloring} (\emph{MVC-coloring} for short), if there is a vertex-monochromatic path joining any two vertices in the graph. For a connected graph $G$, the \emph{monochromatic vertex-connection number}, denoted by $mvc(G)$, is defined to be the maximum number of colors used in an \emph{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. In this paper, we mainly investigate the Erd\H{o}s-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.

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Qingqiong Cai, Xueliang Li, Di Wu. 2015-03-31. Some extremal results on the colorful monochromatic vertex-connectivity of a graph. https://arxiv.org/abs/1503.08941

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