Displaying similar documents to “Set vertex colorings and joins of graphs”

Radio antipodal colorings of graphs

Gary Chartrand, David Erwin, Ping Zhang (2002)

Mathematica Bohemica

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A radio antipodal coloring of a connected graph G with diameter d is an assignment of positive integers to the vertices of G , with x V ( G ) assigned c ( x ) , such that d ( u , v ) + | c ( u ) - c ( v ) | d for every two distinct vertices u , v of G , where d ( u , v ) is the distance between u and v in G . The radio antipodal coloring number a c ( c ) of a radio antipodal coloring c of G is the maximum color assigned to a vertex of G . The radio antipodal chromatic number a c ( G ) of G is min { a c ( c ) } over all radio antipodal colorings c of G . Radio antipodal chromatic numbers...

Defining sets in (proper) vertex colorings of the Cartesian product of a cycle with a complete graph

D. Ali Mojdeh (2006)

Discussiones Mathematicae Graph Theory

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In a given graph G = (V,E), a set of vertices S with an assignment of colors to them is said to be a defining set of the vertex coloring of G, if there exists a unique extension of the colors of S to a c ≥ χ(G) coloring of the vertices of G. A defining set with minimum cardinality is called a minimum defining set and its cardinality is the defining number, denoted by d(G,c). The d(G = Cₘ × Kₙ, χ(G)) has been studied. In this note we show that the exact value of defining number d(G =...

On detectable colorings of graphs

Henry Escuadro, Ping Zhang (2005)

Mathematica Bohemica

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Let G be a connected graph of order n 3 and let c E ( G ) { 1 , 2 , ... , k } be a coloring of the edges of G (where adjacent edges may be colored the same). For each vertex v of G , the color code of v with respect to c is the k -tuple c ( v ) = ( a 1 , a 2 , , a k ) , where a i is the number of edges incident with v that are colored i ( 1 i k ). The coloring c is detectable if distinct vertices have distinct color codes. The detection number det ( G ) of G is the minimum positive integer k for which G has a detectable k -coloring. We establish a formula for the...

On the dominator colorings in trees

Houcine Boumediene Merouane, Mustapha Chellali (2012)

Discussiones Mathematicae Graph Theory

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In a graph G, a vertex is said to dominate itself and all its neighbors. A dominating set of a graph G is a subset of vertices that dominates every vertex of G. The domination number γ(G) is the minimum cardinality of a dominating set of G. A proper coloring of a graph G is a function from the set of vertices of the graph to a set of colors such that any two adjacent vertices have different colors. A dominator coloring of a graph G is a proper coloring such that every vertex of V dominates...

On stratification and domination in graphs

Ralucca Gera, Ping Zhang (2006)

Discussiones Mathematicae Graph Theory

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A graph G is 2-stratified if its vertex set is partitioned into two classes (each of which is a stratum or a color class), where the vertices in one class are colored red and those in the other class are colored blue. Let F be a 2-stratified graph rooted at some blue vertex v. An F-coloring of a graph is a red-blue coloring of the vertices of G in which every blue vertex v belongs to a copy of F rooted at v. The F-domination number γ F ( G ) is the minimum number of red vertices in an F-coloring...