Discussiones Mathematicae Graph Theory (Feb 2018)

Rainbow Vertex-Connection and Forbidden Subgraphs

  • Li Wenjing,
  • Li Xueliang,
  • Zhang Jingshu

DOI
https://doi.org/10.7151/dmgt.2004
Journal volume & issue
Vol. 38, no. 1
pp. 143 – 154

Abstract

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A path in a vertex-colored graph is called vertex-rainbow if its internal vertices have pairwise distinct colors. A vertex-colored graph G is rainbow vertex-connected if for any two distinct vertices of G, there is a vertex-rainbow path connecting them. For a connected graph G, the rainbow vertex-connection number of G, denoted by rvc(G), is defined as the minimum number of colors that are required to make G rainbow vertex-connected. In this paper, we find all the families ℱ of connected graphs with |ℱ| ∈ {1, 2}, for which there is a constant kℱ such that, for every connected ℱ-free graph G, rvc(G) ≤ diam(G) + kℱ, where diam(G) is the diameter of G.

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