Abstract
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.
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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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Supported by NSFC Nos. 11531011 and 11701297.
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Cai, Q., Li, X. & Wu, D. Some extremal results on the colorful monochromatic vertex-connectivity of a graph. J Comb Optim 35, 1300–1311 (2018). https://doi.org/10.1007/s10878-018-0258-x
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DOI: https://doi.org/10.1007/s10878-018-0258-x
Keywords
- Vertex-monochromatic path
- MVC-coloring
- Monochromatic vertex-connection number
- Erdős–Gallai-type problem
- Nordhaus–Gaddum-type problem