Oriented cliques and colorings of graphs with low maximum degree

Dybizbański, Janusz; Ochem, Pascal; Pinlou, Alexandre; Szepietowski, Andrzej

doi:10.1016/j.disc.2020.111829

Cited by 9 publications

(9 citation statements)

References 21 publications

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“…The above-mentioned upper bound v o ðGÞ 16 was later improved by Sopena and Vignal [8] to 11. In recent years Duffy et al [2] proved that 9 colors are enough for members of the family G c , the same bound for G was proved by Dybizbański, Ochem et al [3]. Finally, Duffy [1] proves that v o ðG c Þ 8.…”

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confidence: 76%

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Coloring Graphs in Oriented Coloring of Cubic Graphs

Dybizbański

2022

Graphs and Combinatorics

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Oriented coloring of an oriented graph G is an arc-preserving homomorphism from G into a tournament H. We say that the graph H is universal for a family of oriented graphs $$\mathcal {C}$$ C if for every $$G\in \mathcal {C}$$ G ∈ C there exists a homomorphism from G into H. We are interested in finding a universal graph for the family of orientations of cubic graphs. In this paper we present constructive proof that: if there exists a universal graph H on 7 vertices for every orientation of cubic graphs, then minimum out-degree and minimum in-degree of H are equal to 2. That gives a negative answer to the question presented in Pinlou’s PHD thesis.

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confidence: 76%

“…In summary, the difference between known upper bound and lower bound of v o ðG c Þ equals 1. This motivated us to examine possible coloring graphs on 7 vertices which can color all orientations of graphs with maximum degree 3…”

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confidence: 99%

Coloring Graphs in Oriented Coloring of Cubic Graphs

Dybizbański

2022

Graphs and Combinatorics

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“…Now, for some p, q ≥ 1, we say that #» G has Property P p,q , if for every orientation p-vector S and every vertex p-vector X, there are at least q vertices of #» G that comply with X with respect to S. Back to Paley tournaments, we are interested in the result below. Be aware that [11] is not the first place in which this result ever appeared (see e.g. [6]); however, we advise the reader to refer to [11] because the terminology used there by the authors is closer to ours, and also the proof they provide is fully self-contained.…”

Section: Paley and Tromp Constructionsmentioning

confidence: 92%

“…Be aware that [11] is not the first place in which this result ever appeared (see e.g. [6]); however, we advise the reader to refer to [11] because the terminology used there by the authors is closer to ours, and also the proof they provide is fully self-contained. Theorem 2.7 (Dybizbański, Ochem, Pinlou, Szepietowski [11]).…”

Section: Paley and Tromp Constructionsmentioning

confidence: 92%

On the pushable chromatic number of various types of grids

Bensmail

Das

Lajou

et al. 2023

Discrete Applied Mathematics

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“…For every oriented graph the oriented chromatic number can be bounded (exponentially) by its maximum vertex degree ∆ [KSZ97]. For small vertex degrees ∆ ≤ 7 there are better bounds in [Duf19,DOPS20].…”

Section: Coloring Transitive Acyclic Digraphsmentioning

confidence: 99%

Efficient computation of the oriented chromatic number of recursively defined digraphs

Gurski¹,

Komander²,

Lindemann³

2020

Preprint

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In this paper we consider colorings of oriented graphs, i.e. digraphs without cycles of length 2. Given some oriented graph G = (V, E), an oriented r-coloring for G is a partition of the vertex set V into r independent sets, such that all the arcs between two of these sets have the same direction. The oriented chromatic number of G is the smallest integer r such that G permits an oriented r-coloring. Deciding whether an acyclic digraph has an oriented 4-coloring is NP-hard, which motivates to consider the problem on special graph classes.In this paper we consider the Oriented Chromatic Number problem on classes of recursively defined oriented graphs. Oriented co-graphs (short for oriented complement reducible graphs) can be recursively defined defined from the single vertex graph by applying the disjoint union and order composition. This recursive structure allows to compute an optimal oriented coloring and the oriented chromatic number in linear time. We generalize this result using the concept of perfect orderable graphs. Therefore, we show that for acyclic transitive digraphs every greedy coloring along a topological ordering leads to an optimal oriented coloring. Msp-digraphs (short for minimal series-parallel digraphs) can be defined from the single vertex graph by applying the parallel composition and series composition. We prove an upper bound of 7 for the oriented chromatic number for msp-digraphs and we give an example to show that this is bound best possible. We apply this bound and the recursive structure of msp-digraphs to obtain a linear time solution for computing the oriented chromatic number of msp-digraphs.In order to generalize the results on computing the oriented chromatic number of special graph classes, we consider the parameterized complexity of the Oriented Chromatic Number problem by so-called structural parameters, which are measuring the difficulty of decomposing a graph into a special tree-structure.

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Oriented cliques and colorings of graphs with low maximum degree

Cited by 9 publications

References 21 publications

Coloring Graphs in Oriented Coloring of Cubic Graphs

Coloring Graphs in Oriented Coloring of Cubic Graphs

On the pushable chromatic number of various types of grids

Efficient computation of the oriented chromatic number of recursively defined digraphs

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