On detectable colorings of graphs

Henry Escuadro; Ping Zhang

Displaying similar documents to “On detectable 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 $c V (G) \to ℕ$ 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 $NC (u) \neq NC (v)$ 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 $c E (G) \to {1, 2, ..., k}$ , $k \in ℕ$ , 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,...

Multicolor Ramsey numbers for paths and cycles

Tomasz Dzido (2005)

Discussiones Mathematicae Graph Theory

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For given graphs G₁,G₂,...,Gₖ, k ≥ 2, the multicolor Ramsey number R(G₁,G₂,...,Gₖ) is the smallest integer n such that if we arbitrarily color the edges of the complete graph on n vertices with k colors, then it is always a monochromatic copy of some $G_{i}$ , for 1 ≤ i ≤ k. We give a lower bound for k-color Ramsey number R(Cₘ,Cₘ,...,Cₘ), where m ≥ 8 is even and Cₘ is the cycle on m vertices. In addition, we provide exact values for Ramsey numbers R(P₃,Cₘ,Cₚ), where P₃ is the path on 3 vertices,...

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 =...