A note on graph coloring extensions and list-colorings.
Axenovich, Maria (2003)
The Electronic Journal of Combinatorics [electronic only]
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Axenovich, Maria (2003)
The Electronic Journal of Combinatorics [electronic only]
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Poppy Immel, Paul S. Wenger (2017)
Discussiones Mathematicae Graph Theory
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A distinguishing coloring of a graph G is a coloring of the vertices so that every nontrivial automorphism of G maps some vertex to a vertex with a different color. The distinguishing number of G is the minimum k such that G has a distinguishing coloring where each vertex is assigned a color from {1, . . . , k}. A list assignment to G is an assignment L = {L(v)}v∈V (G) of lists of colors to the vertices of G. A distinguishing L-coloring of G is a distinguishing coloring of G where the...
Axenovich, Maria (2006)
The Electronic Journal of Combinatorics [electronic only]
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Axenovich, Maria, Martin, Ryan (2008)
The Electronic Journal of Combinatorics [electronic only]
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Jean-Sébastien Sereni, Zelealem B. Yilma (2013)
Discussiones Mathematicae Graph Theory
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We provide a tight bound on the set chromatic number of a graph in terms of its chromatic number. Namely, for all graphs G, we show that χs(G) > ⌈log2 χ(G)⌉ + 1, where χs(G) and χ(G) are the set chromatic number and the chromatic number of G, respectively. This answers in the affirmative a conjecture of Gera, Okamoto, Rasmussen and Zhang.
Yuster, Raphael (2006)
The Electronic Journal of Combinatorics [electronic only]
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Wayne Goddard, Kirsti Wash, Honghai Xu (2015)
Discussiones Mathematicae Graph Theory
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Given a coloring of the vertices, we say subgraph H is monochromatic if every vertex of H is assigned the same color, and rainbow if no pair of vertices of H are assigned the same color. Given a graph G and a graph F, we define an F-WORM coloring of G as a coloring of the vertices of G without a rainbow or monochromatic subgraph H isomorphic to F. We present some results on this concept especially as regards to the existence, complexity, and optimization within certain graph classes....
Albertson, Michael O., Chappell, Glenn G., Kierstead, H.A., Kündgen, André, Ramamurthi, Radhika (2004)
The Electronic Journal of Combinatorics [electronic only]
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Alishahi, Meysam, Taherkhani, Ali, Thomassen, Carsten (2011)
The Electronic Journal of Combinatorics [electronic only]
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Alexander Halperin, Colton Magnant, Kyle Pula (2014)
Discussiones Mathematicae Graph Theory
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An edge-colored cycle is rainbow if its edges are colored with distinct colors. A Gallai (multi)graph is a simple, complete, edge-colored (multi)graph lacking rainbow triangles. As has been previously shown for Gallai graphs, we show that Gallai multigraphs admit a simple iterative construction. We then use this structure to prove Ramsey-type results within Gallai colorings. Moreover, we show that Gallai multigraphs give rise to a surprising and highly structured decomposition into directed...
Wayne Goddard, Honghai Xu (2016)
Discussiones Mathematicae Graph Theory
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Given a coloring of the vertices of a graph G, we say a subgraph is rainbow if its vertices receive distinct colors. For a graph F, we define the F-upper chromatic number of G as the maximum number of colors that can be used to color the vertices of G such that there is no rainbow copy of F. We present some results on this parameter for certain graph classes. The focus is on the case that F is a star or triangle. For example, we show that the K3-upper chromatic number of any maximal...
Gary Chartrand, Futaba Okamoto, Craig W. Rasmussen, Ping Zhang (2009)
Discussiones Mathematicae Graph Theory
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For a nontrivial connected graph G, let c: V(G)→ 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) ≠ 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 χₛ(G) of G. The set chromatic numbers of some well-known classes of graphs...
Caro, Yair, Yuster, Raphael (2003)
The Electronic Journal of Combinatorics [electronic only]
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Caro, Yair, Yuster, Raphael (1999)
The Electronic Journal of Combinatorics [electronic only]
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Evelyne Flandrin, Hao Li, Antoni Marczyk, Jean-François Saclé, Mariusz Woźniak (2017)
Discussiones Mathematicae Graph Theory
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A total k-coloring of a graph G is a coloring of vertices and edges of G using colors of the set [k] = {1, . . . , k}. These colors can be used to distinguish the vertices of G. There are many possibilities of such a distinction. In this paper, we consider the sum of colors on incident edges and adjacent vertices.