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Vertex rainbow colorings of graphs

Futaba Fujie-OkamotoKyle KolasinskiJianwei LinPing Zhang — 2012

Discussiones Mathematicae Graph Theory

In a properly vertex-colored graph G, a path P is a rainbow path if no two vertices of P have the same color, except possibly the two end-vertices of P. If every two vertices of G are connected by a rainbow path, then G is vertex rainbow-connected. A proper vertex coloring of a connected graph G that results in a vertex rainbow-connected graph is a vertex rainbow coloring of G. The minimum number of colors needed in a vertex rainbow coloring of G is the vertex rainbow connection number vrc(G) of...

The k -metric colorings of a graph

Futaba Fujie-OkamotoWillem RenzemaPing Zhang — 2012

Mathematica Bohemica

For a nontrivial connected graph G of order n , the detour distance D ( u , v ) between two vertices u and v in G is the length of a longest u - v path in G . Detour distance is a metric on the vertex set of G . For each integer k with 1 k n - 1 , a coloring c : V ( G ) is a k -metric coloring of G if | c ( u ) - c ( v ) | + D ( u , v ) k + 1 for every two distinct vertices u and v of G . The value χ m k ( c ) of a k -metric coloring c is the maximum color assigned by c to a vertex of G and the k -metric chromatic number χ m k ( G ) of G is the minimum value of a k -metric coloring of G . For every...

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