Displaying similar documents to “Generalized circular colouring of graphs”

Equitable coloring of Kneser graphs

Robert Fidytek, Hanna Furmańczyk, Paweł Żyliński (2009)

Discussiones Mathematicae Graph Theory

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The Kneser graph K(n,k) is the graph whose vertices correspond to k-element subsets of set {1,2,...,n} and two vertices are adjacent if and only if they represent disjoint subsets. In this paper we study the problem of equitable coloring of Kneser graphs, namely, we establish the equitable chromatic number for graphs K(n,2) and K(n,3). In addition, for sufficiently large n, a tight upper bound on equitable chromatic number of graph K(n,k) is given. Finally, the cases of K(2k,k) and K(2k+1,k)...

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...

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.

Some applications of pq-groups in graph theory

Geoffrey Exoo (2004)

Discussiones Mathematicae Graph Theory

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We describe some new applications of nonabelian pq-groups to construction problems in Graph Theory. The constructions include the smallest known trivalent graph of girth 17, the smallest known regular graphs of girth five for several degrees, along with four edge colorings of complete graphs that improve lower bounds on classical Ramsey numbers.

Simplicial and nonsimplicial complete subgraphs

Terry A. McKee (2011)

Discussiones Mathematicae Graph Theory

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Define a complete subgraph Q to be simplicial in a graph G when Q is contained in exactly one maximal complete subgraph ('maxclique') of G; otherwise, Q is nonsimplicial. Several graph classes-including strong p-Helly graphs and strongly chordal graphs-are shown to have pairs of peculiarly related new characterizations: (i) for every k ≤ 2, a certain property holds for the complete subgraphs that are in k or more maxcliques of G, and (ii) in every induced subgraph H of G, that...

4-chromatic Koester graphs

Andrey A. Dobrynin, Leonid S. Mel'nikov (2012)

Discussiones Mathematicae Graph Theory

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Let G be a simple 4-regular plane graph and let S be a decomposition of G into edge-disjoint cycles. Suppose that every two adjacent edges on a face belong to different cycles of S. Such a graph G arises as a superposition of simple closed curves in the plane with tangencies disallowed. Studies of coloring of graphs of this kind were originated by Grötzsch. Two 4-chromatic graphs generated by circles in the plane were constructed by Koester in 1984 [10,11,12]. Until now, no other examples...

Equitable Colorings Of Corona Multiproducts Of Graphs

Hanna Furmánczyk, Marek Kubale, Vahan V. Mkrtchyan (2017)

Discussiones Mathematicae Graph Theory

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A graph is equitably k-colorable if its vertices can be partitioned into k independent sets in such a way that the numbers of vertices in any two sets differ by at most one. The smallest k for which such a coloring exists is known as the equitable chromatic number of G and denoted by 𝜒=(G). It is known that the problem of computation of 𝜒=(G) is NP-hard in general and remains so for corona graphs. In this paper we consider the same model of coloring in the case of corona multiproducts...

On 1-dependent ramsey numbers for graphs

E.J. Cockayne, C.M. Mynhardt (1999)

Discussiones Mathematicae Graph Theory

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A set X of vertices of a graph G is said to be 1-dependent if the subgraph of G induced by X has maximum degree one. The 1-dependent Ramsey number t₁(l,m) is the smallest integer n such that for any 2-edge colouring (R,B) of Kₙ, the spanning subgraph B of Kₙ has a 1-dependent set of size l or the subgraph R has a 1-dependent set of size m. The 2-edge colouring (R,B) is a t₁(l,m) Ramsey colouring of Kₙ if B (R, respectively) does not contain a 1-dependent set of size l (m, respectively);...

Bipartite graphs that are not circle graphs

André Bouchet (1999)

Annales de l'institut Fourier

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The following result is proved: if a bipartite graph is not a circle graph, then its complement is not a circle graph. The proof uses Naji’s characterization of circle graphs by means of a linear system of equations with unknowns in GF ( 2 ) . At the end of this short note I briefly recall the work of François Jaeger on circle graphs.

α-Labelings of a Class of Generalized Petersen Graphs

Anna Benini, Anita Pasotti (2015)

Discussiones Mathematicae Graph Theory

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An α-labeling of a bipartite graph Γ of size e is an injective function f : V (Γ) → {0, 1, 2, . . . , e} such that {|ƒ(x) − ƒ(y)| : [x, y] ∈ E(Γ)} = {1, 2, . . . , e} and with the property that its maximum value on one of the two bipartite sets does not reach its minimum on the other one. We prove that the generalized Petersen graph PSn,3 admits an α-labeling for any integer n ≥ 1 confirming that the conjecture posed by Vietri in [10] is true. In such a way we obtain an infinite class...

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.