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A Note on the Total Detection Numbers of Cycles

Henry E. EscuadroFutaba FujieChad E. Musick — 2015

Discussiones Mathematicae Graph Theory

Let G be a connected graph of size at least 2 and c :E(G)→{0, 1, . . . , k− 1} an edge coloring (or labeling) of G using k labels, where adjacent edges may be assigned the same label. For each vertex v of G, the color code of v with respect to c is the k-vector code(v) = (a0, a1, . . . , ak−1), where ai is the number of edges incident with v that are labeled i for 0 ≤ i ≤ k − 1. The labeling c is called a detectable labeling if distinct vertices in G have distinct color codes. The value val(c) of...

On Monochromatic Subgraphs of Edge-Colored Complete Graphs

Eric AndrewsFutaba FujieKyle KolasinskiChira LumduanhomAdam Yusko — 2014

Discussiones Mathematicae Graph Theory

In a red-blue coloring of a nonempty graph, every edge is colored red or blue. If the resulting edge-colored graph contains a nonempty subgraph G without isolated vertices every edge of which is colored the same, then G is said to be monochromatic. For two nonempty graphs G and H without isolated vertices, the mono- chromatic Ramsey number mr(G,H) of G and H is the minimum integer n such that every red-blue coloring of Kn results in a monochromatic G or a monochromatic H. Thus, the standard Ramsey...

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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