Two results on the digraph chromatic number

Harutyunyan, Ararat; Mohar, Bojan

doi:10.1016/j.disc.2012.01.028

Cited by 34 publications

(32 citation statements)

References 17 publications

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“…For example, under this definition, results on Gallai colourings and list colourings also generalize to digraphs [9]. And as proved by Harutyunyan and Mohar [10], there exist digraphs of maximum degree ∆ and of arbitrarily large digirth whose chromatic number is at least c∆ log ∆ , thus generalizing a result of Bollobás [4] for undirected graphs. Finally, there is a fascinating connection [18] between χ(D) and its spectral radius (i.e., the largest modulus of an eigenvalue of the adjacency matrix A(D)), thus showing that the relationship between the chromatic number of a graph and its eigenvalues [21] also applies to digraphs.…”

Section: The Chromatic Number Of Digraphsmentioning

confidence: 83%

The edge density of critical digraphs

Hoshino

Kawarabayashi

2014

Combinatorica

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Let χ(G) denote the chromatic number of a graph G. We say that G is k-critical if χ(G) = k and χ(H) < k for every proper subgraph H ⊂ G. Over the years, many properties of kcritical graphs have been discovered, including improved upper and lower bounds for ||G||, the number of edges in a k-critical graph, as a function of |G|, the number of vertices. In this note, we analyze this edge density problem for directed graphs, where the chromatic number χ(D) of a digraph D is defined to be the fewest number of colours needed to colour the vertices of D so that each colour class induces an acyclic subgraph. For each k ≥ 3, we construct an infinite family of sparse k-critical digraphs for which ||D|| < ( k 2 −k+1 2 )|D| and an infinite family of dense k-critical digraphs for which ||D|| > 1 2 − 1 2 k−1 |D| 2 . One corollary of our results is an explicit construction of an infinite family of k-critical digraphs of digirth l, for any pair of integers k, l ≥ 3. This extends a result by Bokal et al. who used a probabilistic approach to demonstrate the existence of one such digraph.

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Section: The Chromatic Number Of Digraphsmentioning

confidence: 83%

The edge density of critical digraphs

Hoshino

Kawarabayashi

2014

Combinatorica

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“…A k-coloring of a digraph D is a partition of V (D) into k acyclic sets, and the chromatic number χ(D) is the minimum number k for which D admits a k-coloring. This digraph invariant was introduced by Neumann-Lara [13], and naturally generalizes many results on the graph chromatic number (see, for example, [3], [9] [10], [11], [12]).…”

Section: Introductionmentioning

confidence: 82%

Coloring Dense Digraphs

2019

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The chromatic number of a digraph D is the minimum number of acyclic subgraphs covering the vertex set of D. A tournament H is a hero if every H-free tournament T has chromatic number bounded by a function of H. Inspired by the celebrated Erdős-Hajnal conjecture, Berger et al. fully characterized the class of heroes in 2013. Motivated by a question of the first author and Colin McDiarmid, we study digraphs which we call superheroes. A digraph H is a superhero if every H-free digraph D has chromatic number bounded by a function of H and α(D), the independence number of the underlying graph of D. We prove here that a digraph is a superhero if and only if it is a hero, and hence characterize all superheroes. This answers a question of Pierre Aboulker.

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“…Then (18) a ≺ b and |P | ≡ f (b) − f (a) + 1 (mod k). By (11), at least one of a and b is in P i+1 \D i . Depending on the locations of a and b, we distinguish among three cases.…”

Section: Acyclic Coloringmentioning

confidence: 99%

Coloring digraphs with forbidden cycles

Chen

Zang

2015

Journal of Combinatorial Theory, Series B

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Let k and r be two integers with k ≥ 2 and k ≥ r ≥ 1. In this paper we show that (1) if a strongly connected digraph D contains no directed cycle of length 1 modulo k, then D is k-colorable; and (2) if a digraph D contains no directed cycle of length r modulo k, then D can be vertex-colored with k colors so that each color class induces an acyclic subdigraph in D. The first result gives an affirmative answer to a question posed by Tuza in 1992, and the second implies the following strong form of a conjecture of Diwan, Kenkre and Vishwanathan: If an undirected graph G contains no cycle of length r modulo k, then G is k-colorable if r = 2 and (k + 1)-colorable otherwise. Our results also strengthen several classical theorems on graph coloring proved

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Two results on the digraph chromatic number

Cited by 34 publications

References 17 publications

The edge density of critical digraphs

The edge density of critical digraphs

Coloring Dense Digraphs

Coloring digraphs with forbidden cycles

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