Gallai's Theorem for List Coloring of Digraphs

Harutyunyan, Ararat; Mohar, Bojan

doi:10.1137/100803870

Cited by 29 publications

(33 citation statements)

References 9 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.…”

Section: The Chromatic Number Of Digraphsmentioning

confidence: 68%

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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: 68%

“…THE EDGE DENSITY OF CRITICAL DIGRAPHS 9 We determine the following inequality, which holds true for any k, n, l ≥ 3:…”

mentioning

confidence: 99%

The edge density of critical digraphs

Hoshino

Kawarabayashi

2014

Combinatorica

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“…Case 1. a, b ∈ P i+1 . By (8), (9) and (10), we have a = u p and b = u q for some p and (10). It follows that Q 1 ∈ P s+1 if P i+1 ∈ P s and that Q 1 ∈ Q s+1 if P i+1 ∈ Q s , contradicting (7) in either subcase.…”

Section: Acyclic Coloringmentioning

confidence: 92%

“…It follows that Q 1 ∈ P s+1 if P i+1 ∈ P s and that Q 1 ∈ Q s+1 if P i+1 ∈ Q s , contradicting (7) in either subcase. If b = u h , then u 0 = u h by (9) and (10). Thus Q 1 is a forward D i -ear from (10), and hence Q 1 ∈ P 1 , contradicting (5).…”

Section: Acyclic Coloringmentioning

confidence: 95%

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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“…The list dichromatic number ⃗ ( ) of is the smallest integer such that is -colorable for any -list-assignment . We note that the definition of list coloring of digraphs is not quite new as it first appeared in [11] where the authors derived an analog of Gallai's theorem for digraphs, as well as in [13]. Our goal in this article is to initiate the study of the list dichromatic number.…”

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confidence: 99%

List coloring digraphs

Bensmail

Harutyunyan

2017

Journal of Graph Theory

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The dichromatic number χ⃗false(Dfalse) of a digraph D is the least number k such that the vertex set of D can be partitioned into k parts each of which induces an acyclic subdigraph. Introduced by Neumann‐Lara in 1982, this digraph invariant shares many properties with the usual chromatic number of graphs and can be seen as the natural analog of the graph chromatic number. In this article, we study the list dichromatic number of digraphs, giving evidence that this notion generalizes the list chromatic number of graphs. We first prove that the list dichromatic number and the dichromatic number behave the same in many contexts, such as in small digraphs (by proving a directed version of Ohba's conjecture), tournaments, and random digraphs. We then consider bipartite digraphs, and show that their list dichromatic number can be as large as Ω(prefixlog2n). We finally give a Brooks‐type upper bound on the list dichromatic number of digon‐free digraphs.

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Gallai's Theorem for List Coloring of Digraphs

Cited by 29 publications

References 9 publications

The edge density of critical digraphs

The edge density of critical digraphs

Coloring digraphs with forbidden cycles

List coloring digraphs

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