Displaying similar documents to “Kaleidoscopic Colorings of Graphs”

Set vertex colorings and joins of graphs

Futaba Okamoto, Craig W. Rasmussen, Ping Zhang (2009)

Czechoslovak Mathematical Journal

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For a nontrivial connected graph G , let c V ( G ) be a vertex coloring of G where adjacent vertices may be colored the same. For a vertex v of G , the neighborhood color set NC ( v ) is the set of colors of the neighbors of v . The coloring c is called a set coloring if NC ( u ) NC ( v ) for every pair u , v of adjacent vertices of G . The minimum number of colors required of such a coloring is called the set chromatic number χ s ( G ) . A study is made of the set chromatic number of the join G + H of two graphs G and H . Sharp lower...

Rainbow connection in graphs

Gary Chartrand, Garry L. Johns, Kathleen A. McKeon, Ping Zhang (2008)

Mathematica Bohemica

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Let G be a nontrivial connected graph on which is defined a coloring c E ( G ) { 1 , 2 , ... , k } , k , of the edges of G , where adjacent edges may be colored the same. A path P in G is a rainbow path if no two edges of P are colored the same. The graph G is rainbow-connected if G contains a rainbow u - v path for every two vertices u and v of G . The minimum k for which there exists such a k -edge coloring is the rainbow connection number r c ( G ) of G . If for every pair u , v of distinct vertices, G contains a rainbow u - v geodesic,...

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...

Maximum Edge-Colorings Of Graphs

Stanislav Jendrol’, Michaela Vrbjarová (2016)

Discussiones Mathematicae Graph Theory

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An r-maximum k-edge-coloring of G is a k-edge-coloring of G having a property that for every vertex v of degree dG(v) = d, d ≥ r, the maximum color, that is present at vertex v, occurs at v exactly r times. The r-maximum index [...] χr′(G) χ r ' ( G ) is defined to be the minimum number k of colors needed for an r-maximum k-edge-coloring of graph G. In this paper we show that [...] χr′(G)≤3 χ r ' ( G ) 3 for any nontrivial connected graph G and r = 1 or 2. The bound 3 is tight. All graphs G with [...] χ1′(G)=i...

Radio k-colorings of paths

Gary Chartrand, Ladislav Nebeský, Ping Zhang (2004)

Discussiones Mathematicae Graph Theory

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For a connected graph G of diameter d and an integer k with 1 ≤ k ≤ d, a radio k-coloring of G is an assignment c of colors (positive integers) to the vertices of G such that d(u,v) + |c(u)- c(v)| ≥ 1 + k for every two distinct vertices u and v of G, where d(u,v) is the distance between u and v. The value rcₖ(c) of a radio k-coloring c of G is the maximum color assigned to a vertex of G. The radio k-chromatic number rcₖ(G) of G is the minimum value of rcₖ(c) taken over all radio k-colorings...