Rainbow numbers for small stars with one edge added

Izolda Gorgol; Ewa Łazuka

Displaying similar documents to “Rainbow numbers for small stars with one edge added”

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...

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,...

Kaleidoscopic Colorings of Graphs

Gary Chartrand, Sean English, Ping Zhang (2017)

Discussiones Mathematicae Graph Theory

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For an r-regular graph G, let c : E(G) → [k] = 1, 2, . . . , k, k ≥ 3, be an edge coloring of G, where every vertex of G is incident with at least one edge of each color. For a vertex v of G, the multiset-color cm(v) of v is defined as the ordered k-tuple (a1, a2, . . . , ak) or a1a2 … ak, where ai (1 ≤ i ≤ k) is the number of edges in G colored i that are incident with v. The edge coloring c is called k-kaleidoscopic if cm(u) ≠ cm(v) for every two distinct vertices u and v of G. A regular...

Hajós' theorem for list colorings of hypergraphs

Claude Benzaken, Sylvain Gravier, Riste Skrekovski (2003)

Discussiones Mathematicae Graph Theory

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A well-known theorem of Hajós claims that every graph with chromathic number greater than k can be constructed from disjoint copies of the complete graph $K_{k + 1}$ by repeated application of three simple operations. This classical result has been extended in 1978 to colorings of hypergraphs by C. Benzaken and in 1996 to list-colorings of graphs by S. Gravier. In this note, we capture both variations to extend Hajós’ theorem to list-colorings of hypergraphs.

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,...

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...

A rainbow $k$ -matching in the complete graph with $r$ colors.

Fujita, Shinya, Kaneko, Atsushi, Schiermeyer, Ingo, Suzuki, Kazuhiro (2009)

The Electronic Journal of Combinatorics [electronic only]

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Recognizable colorings of cycles and trees

Michael J. Dorfling, Samantha Dorfling (2012)

Discussiones Mathematicae Graph Theory

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For a graph G and a vertex-coloring c:V(G) → 1,2, ...,k, the color code of a vertex v is the (k+1)-tuple (a₀,a₁, ...,aₖ), where a₀ = c(v), and for 1 ≤ i ≤ k, $a_{i}$ is the number of neighbors of v colored i. A recognizable coloring is a coloring such that distinct vertices have distinct color codes. The recognition number of a graph is the minimum k for which G has a recognizable k-coloring. In this paper we prove three conjectures of Chartrand et al. in [8] regarding the recognition number...

Three edge-coloring conjectures

Richard H. Schelp (2002)

Discussiones Mathematicae Graph Theory

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The focus of this article is on three of the author's open conjectures. The article itself surveys results relating to the conjectures and shows where the conjectures are known to hold.

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 k-intersection edge colourings

Rahul Muthu, N. Narayanan, C.R. Subramanian (2009)

Discussiones Mathematicae Graph Theory

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We propose the following problem. For some k ≥ 1, a graph G is to be properly edge coloured such that any two adjacent vertices share at most k colours. We call this the k-intersection edge colouring. The minimum number of colours sufficient to guarantee such a colouring is the k-intersection chromatic index and is denoted χ’ₖ(G). Let fₖ be defined by $f ₖ (Δ) = m a x_{G : Δ (G) = Δ} χ^{'} ₖ (G)$ . We show that fₖ(Δ) = Θ(Δ²/k). We also discuss some open problems.

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...