Displaying similar documents to “The n-minimal chromatic multiplicity of a graph”

Solutions of Some L(2, 1)-Coloring Related Open Problems

Nibedita Mandal, Pratima Panigrahi (2016)

Discussiones Mathematicae Graph Theory

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An L(2, 1)-coloring (or labeling) of a graph G is a vertex coloring f : V (G) → Z+ ∪ {0} such that |f(u) − f(v)| ≥ 2 for all edges uv of G, and |f(u)−f(v)| ≥ 1 if d(u, v) = 2, where d(u, v) is the distance between vertices u and v in G. The span of an L(2, 1)-coloring is the maximum color (or label) assigned by it. The span of a graph G is the smallest integer λ such that there exists an L(2, 1)-coloring of G with span λ. An L(2, 1)-coloring of a graph with span equal to the span of...

Generalized circular colouring of graphs

Peter Mihók, Janka Oravcová, Roman Soták (2011)

Discussiones Mathematicae Graph Theory

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Let P be a graph property and r,s ∈ N, r ≥ s. A strong circular (P,r,s)-colouring of a graph G is an assignment f:V(G) → {0,1,...,r-1}, such that the edges uv ∈ E(G) satisfying |f(u)-f(v)| < s or |f(u)-f(v)| > r - s, induce a subgraph of G with the propery P. In this paper we present some basic results on strong circular (P,r,s)-colourings. We introduce the strong circular P-chromatic number of a graph and we determine the strong circular P-chromatic number of complete graphs for...

2-Tone Colorings in Graph Products

Jennifer Loe, Danielle Middelbrooks, Ashley Morris, Kirsti Wash (2015)

Discussiones Mathematicae Graph Theory

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A variation of graph coloring known as a t-tone k-coloring assigns a set of t colors to each vertex of a graph from the set {1, . . . , k}, where the sets of colors assigned to any two vertices distance d apart share fewer than d colors in common. The minimum integer k such that a graph G has a t- tone k-coloring is known as the t-tone chromatic number. We study the 2-tone chromatic number in three different graph products. In particular, given graphs G and H, we bound the 2-tone chromatic...

Pₘ-saturated bipartite graphs with minimum size

Aneta Dudek, A. Paweł Wojda (2004)

Discussiones Mathematicae Graph Theory

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A graph G is said to be H-saturated if G is H-free i.e., (G has no subgraph isomorphic to H) and adding any new edge to G creates a copy of H in G. In 1986 L. Kászonyi and Zs. Tuza considered the following problem: for given m and n find the minimum size sat(n;Pₘ) of Pₘ-saturated graph of order n. They gave the number sat(n;Pₘ) for n big enough. We deal with similar problem for bipartite graphs.

The list linear arboricity of planar graphs

Xinhui An, Baoyindureng Wu (2009)

Discussiones Mathematicae Graph Theory

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The linear arboricity la(G) of a graph G is the minimum number of linear forests which partition the edges of G. An and Wu introduce the notion of list linear arboricity lla(G) of a graph G and conjecture that lla(G) = la(G) for any graph G. We confirm that this conjecture is true for any planar graph having Δ ≥ 13, or for any planar graph with Δ ≥ 7 and without i-cycles for some i ∈ {3,4,5}. We also prove that ⌈½Δ(G)⌉ ≤ lla(G) ≤ ⌈½(Δ(G)+1)⌉ for any planar graph having Δ ≥ 9. ...

Generalized colorings and avoidable orientations

Jenő Szigeti, Zsolt Tuza (1997)

Discussiones Mathematicae Graph Theory

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Gallai and Roy proved that a graph is k-colorable if and only if it has an orientation without directed paths of length k. We initiate the study of analogous characterizations for the existence of generalized graph colorings, where each color class induces a subgraph satisfying a given (hereditary) property. It is shown that a graph is partitionable into at most k independent sets and one induced matching if and only if it admits an orientation containing no subdigraph from a family...

Vertex coloring the square of outerplanar graphs of low degree

Geir Agnarsson, Magnús M. Halldórsson (2010)

Discussiones Mathematicae Graph Theory

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Vertex colorings of the square of an outerplanar graph have received a lot of attention recently. In this article we prove that the chromatic number of the square of an outerplanar graph of maximum degree Δ = 6 is 7. The optimal upper bound for the chromatic number of the square of an outerplanar graph of maximum degree Δ ≠ 6 is known. Hence, this mentioned chromatic number of 7 is the last and only unknown upper bound of the chromatic number in terms of Δ.