Multicolor Ramsey numbers for paths and cycles

Tomasz Dzido

Displaying similar documents to “Multicolor Ramsey numbers for paths and cycles”

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...

On detectable colorings of graphs

Henry Escuadro, Ping Zhang (2005)

Mathematica Bohemica

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Let $G$ be a connected graph of order $n \geq 3$ and let $c E (G) \to {1, 2, ..., k}$ 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 $c (v) = (a_{1}, a_{2}, \dots, a_{k})$ , where $a_{i}$ is the number of edges incident with $v$ that are colored $i$ ( $1 \leq i \leq k$ ). 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...

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,...

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...

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 =...

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 $χ_{r}^{'} (G) \leq 3$ for any nontrivial connected graph G and r = 1 or 2. The bound 3 is tight. All graphs G with [...] χ1′(G)=i...

3-consecutive c-colorings of graphs

Csilla Bujtás, E. Sampathkumar, Zsolt Tuza, M.S. Subramanya, Charles Dominic (2010)

Discussiones Mathematicae Graph Theory

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A 3-consecutive C-coloring of a graph G = (V,E) is a mapping φ:V → ℕ such that every path on three vertices has at most two colors. We prove general estimates on the maximum number ${(χ ̅)}_{3 C C} (G)$ of colors in a 3-consecutive C-coloring of G, and characterize the structure of connected graphs with ${(χ ̅)}_{3 C C} (G) \geq k$ for k = 3 and k = 4.

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 $x \in V (G)$ assigned $c (x)$ , such that $d (u, v) + | c (u) - c (v) | \geq d$ 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...