Coloring digraphs with forbidden cycles

Chen, Zhibin; Ma, Jie; Zang, Wenan

doi:10.1016/j.jctb.2015.06.001

Cited by 19 publications

(23 citation statements)

References 21 publications

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“…The restriction on the cycles lengths in Theorems 3 and 4 provide better bounds. In Theorem 3 whenever s < ⌊ k 2 ⌋, χ A (D) < k and in Theorem 4 whenever k = sp with s ≥ 3, χ A (D) ≤ s < k. Our results are different from those obtained by Chen et al (2015). However, these two cases lower their bound.…”

Section: Introductioncontrasting

confidence: 86%

“…Concerning acyclic colorings, Neumann-Lara (1982) proved that for any fixed integers k and r with k ≥ r ≥ 2, whenever a digraph D contains no cycle of length 0 or 1 modulo r, then χ A (D) ≤ k. In this direction, Chen et al (2015) gave a more general result. They proved for integers k and r with k ≥ 2 and k ≥ r ≥ 1 that: i) if a digraph D contains no cycle of length 1 modulo k, then D can be colored with k colors so that each color class is a stable set; ii) if a digraph D contains no cycle of length r modulo k, then D can be colored with k colors so that each color class induces an acyclic subdigraph of D; iii) 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.…”

Section: Introductionmentioning

confidence: 99%

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New Bounds for the Dichromatic Number of a Digraph

Cordero-Michel,

Galeana-Sánchez

2018

Preprint

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The chromatic number of a graph G, denoted by χ(G), is the minimum k such that G admits a k-coloring of its vertex set in such a way that each color class is an independent set (a set of pairwise nonadjacent vertices).The dichromatic number of a digraph D, denoted by χA(D), is the minimum k such that D admits a k-coloring of its vertex set in such a way that each color class is acyclic.In 1976, Bondy proved that the chromatic number of a digraph D is at most its circumference, the length of a longest cycle.Given a digraph D, we will construct three different graphs whose chromatic numbers bound χA(D). Moreover, we prove: i) for integers k ≥ 2, s ≥ 1 and r1, . . . , rs with k ≥ ri ≥ 0 and ri = 1 for each i ∈ [s], that if all cycles in D have length r modulo k for some r ∈ {r1, . . . , rs}, then χA(D) ≤ 2s + 1; ii) if D has girth g and there are integers k and p, with k ≥ g − 1 ≥ p ≥ 1 such that D contains no cycle of length r modulo ⌈ k p ⌉p for each r ∈ {−p + 2, . . . , 0, . . . , p}, then χA(D) ≤ ⌈ k p ⌉; iii) if D has girth g, the length of a shortest cycle, and circumference c, then χA(D) ≤ ⌈ c−1 g−1 ⌉ + 1, which improves, substantially, the bound proposed by Bondy. Our results show that if we have more information about the lengths of cycles in a digraph, then we can improve the bounds for the dichromatic number known until now.

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Section: Introductioncontrasting

confidence: 86%

Section: Introductionmentioning

confidence: 99%

New Bounds for the Dichromatic Number of a Digraph

Cordero-Michel,

Galeana-Sánchez

2018

Preprint

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“…More generally, one could investigate the minimum number cycles of length r mod k in a graph of chromatic number at least, say, f (r, k). With regards to the existence of such cycles, Chen, Ma and Zang [9] proved that any graph with chromatic number greater than k must contain a cycle of length r mod k for r ∈ {0, . .…”

mentioning

confidence: 99%

A dichotomy theorem for circular colouring reconfiguration

Brewster

McGuinness

Moore

et al. 2016

Theoretical Computer Science

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Let p and q be positive integers with p/q ≥ 2. The "reconfiguration problem" for circular colourings asks, given two (p, q)-colourings f and g of a graph G, is it possible to transform f into g by changing the colour of one vertex at a time such that every intermediate mapping is a (p, q)-colouring? We show that this problem can be solved in polynomial time for 2 ≤ p/q < 4 and that it is PSPACE-complete for p/q ≥ 4. This generalizes a known dichotomy theorem for reconfiguring classical graph colourings. As an application of the reconfiguration algorithm, we show that graphs with fewer than (k − 1)!/2 cycles of length divisible by k are k-colourable.

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“…Diwan, Kenkre and Vishwanathan [13] conjectured that for every pair of integers m and k, if graph G has no cycle of length m modulo k, then the chromatic number of G is at most k + o(k). This was resolved by Chen, Ma and Zang in a recent paper [8], where they also studied the relations between cycle lengths modulo k and chromatic number of digraphs.…”

Section: Cycles Of Consecutive Lengths and Chromatic Numbermentioning

confidence: 99%

Cycle lengths and minimum degree of graphs

Liu

2018

Journal of Combinatorial Theory, Series B

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There has been extensive research on cycle lengths in graphs with large minimum degree. In this paper, we obtain several new and tight results in this area. Let G be a graph with minimum degree at least k + 1. We prove that if G is bipartite, then there are k cycles in G whose lengths form an arithmetic progression with common difference two. For general graph G, we show that G contains ⌊k/2⌋ cycles with consecutive even lengths and k − 3 cycles whose lengths form an arithmetic progression with common difference one or two. In addition, if G is 2-connected and non-bipartite, then G contains ⌊k/2⌋ cycles with consecutive odd lengths.Thomassen (1983) made two conjectures on cycle lengths modulo a fixed integer k: (1) every graph with minimum degree at least k + 1 contains cycles of all even lengths modulo k; (2) every 2-connected non-bipartite graph with minimum degree at least k + 1 contains cycles of all lengths modulo k. These two conjectures, if true, are best possible. Our results confirm both conjectures when k is even. And when k is odd, we show that minimum degree at least k + 4 suffices. This improves all previous results in this direction. Moreover, our results derive new upper bounds of the chromatic number in terms of the longest sequence of cycles with consecutive (even or odd) lengths.

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Coloring digraphs with forbidden cycles

Cited by 19 publications

References 21 publications

New Bounds for the Dichromatic Number of a Digraph

New Bounds for the Dichromatic Number of a Digraph

A dichotomy theorem for circular colouring reconfiguration

Cycle lengths and minimum degree of graphs

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