Single‐conflict colouring

Dvořák, Zdeněk; Kang, Ross J.; Ozeki, Kenta

doi:10.1002/jgt.22646

Cited by 14 publications

(22 citation statements)

References 15 publications

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“…This sufficient condition for Problem 3 is related to independent transversals in hypergraphs, cf. [8,11], and to singleconflict chromatic number [7]. In the most asymmetric settings (when k A and Δ A are fixed constants), condition (i) is sharp up to a constant factor.…”

Section: Theorem 4 Let the Positive Integersmentioning

confidence: 99%

Asymmetric List Sizes in Bipartite Graphs

2021

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Given a bipartite graph with parts A and B having maximum degrees at most $$\Delta _A$$ Δ A and $$\Delta _B$$ Δ B , respectively, consider a list assignment such that every vertex in A or B is given a list of colours of size $$k_A$$ k A or $$k_B$$ k B , respectively. We prove some general sufficient conditions in terms of $$\Delta _A$$ Δ A , $$\Delta _B$$ Δ B , $$k_A$$ k A , $$k_B$$ k B to be guaranteed a proper colouring such that each vertex is coloured using only a colour from its list. These are asymptotically nearly sharp in the very asymmetric cases. We establish one sufficient condition in particular, where $$\Delta _A=\Delta _B=\Delta $$ Δ A = Δ B = Δ , $$k_A=\log \Delta $$ k A = log Δ and $$k_B=(1+o(1))\Delta /\log \Delta $$ k B = ( 1 + o ( 1 ) ) Δ / log Δ as $$\Delta \rightarrow \infty $$ Δ → ∞ . This amounts to partial progress towards a conjecture from 1998 of Krivelevich and the first author. We also derive some necessary conditions through an intriguing connection between the complete case and the extremal size of approximate Steiner systems. We show that for complete bipartite graphs these conditions are asymptotically nearly sharp in a large part of the parameter space. This has provoked the following. In the setup above, we conjecture that a proper list colouring is always guaranteed if $$k_A \ge \Delta _A^\varepsilon $$ k A ≥ Δ A ε and $$k_B \ge \Delta _B^\varepsilon $$ k B ≥ Δ B ε for any $$\varepsilon >0$$ ε > 0 provided $$\Delta _A$$ Δ A and $$\Delta _B$$ Δ B are large enough; if $$k_A \ge C \log \Delta _B$$ k A ≥ C log Δ B and $$k_B \ge C \log \Delta _A$$ k B ≥ C log Δ A for some absolute constant $$C>1$$ C > 1 ; or if $$\Delta _A=\Delta _B = \Delta $$ Δ A = Δ B = Δ and $$ k_B \ge C (\Delta /\log \Delta )^{1/k_A}\log \Delta $$ k B ≥ C ( Δ / log Δ ) 1 / k A log Δ for some absolute constant $$C>0$$ C > 0 . These are asymmetric generalisations of the above-mentioned conjecture of Krivelevich and the first author, and if true are close to best possible. Our general sufficient conditions provide partial progress towards these conjectures.

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Section: Theorem 4 Let the Positive Integersmentioning

confidence: 99%

Asymmetric List Sizes in Bipartite Graphs

2021

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“…Background. The concept of a single-conflict coloring of a graph was first introduced by Dvořák and Postle [4], and independently by Fraigniaud, Heinrich and Kosowski [5], and the notion of the single-conflict chromatic number was later introduced by Dvořák, Esperet, Kang, and Ozeki [3]. In [3], the authors prove the following theorems:…”

Section: Introductionmentioning

confidence: 99%

“…Furthermore, the authors show that a graph of average degree d has a single-conflict chromatic number of at least d log d . One of the main tools used in [3] is the Lovász Local Lemma: Theorem 1.3. [11] Let B be a set of bad events.…”

Section: Introductionmentioning

confidence: 99%

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Single-conflict colorings of degenerate graphs

Bradshaw¹,

Masařík²

2021

Preprint

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We consider single-conflict colorings, a variant of graph colorings in which each edge of a graph has a single forbidden color pair. We show that for any assignment of forbidden color pairs to the edges of a d-degenerate graph G on n vertices of edge-multiplicity at most log log n, O( √ d log n) colors are always enough to color the vertices of G in a way that avoids every forbidden color pair. This answers a question of Dvořák, Esperet, Kang, and Ozeki for simple graphs (Journal of Graph Theory 2021).

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“…This is of course the same problem we consider here, the only important difference being that in our case the pairs in M e are instead included in our graph as parallel edges between u and v, and therefore they contribute to the maximum degree. Our setting most closely resembles that of Dvořák, Esperet, Kang and Ozeki [5], and we use the same notion of conflict degree (we define this in a moment) seen in [13].…”

Section: Introductionmentioning

confidence: 99%

Adaptable and conflict colouring multigraphs with no cycles of length three or four

Jurgen¹,

Molloy²

2021

Preprint

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The adaptable choosability of a multigraph G, denoted ch a (G), is the smallest integer k such that any edge labelling, τ , of G and any assignment of lists of size k to the vertices of G permits a list colouring, σ, of G such that there is no edge e = uv where τ (e) = σ(u) = σ(v).Here we show that for a multigraph G with maximum degree ∆ and no cycles of length 3 or 4, ch a (G) ≤ (2 √ 2 + o(1)) ∆/ ln ∆. Under natural restrictions we can show that the same bound holds for the conflict choosability of G, which is a closely related parameter recently defined by Dvořák, Esperet, Kang and Ozeki.

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Single‐conflict colouring

Cited by 14 publications

References 15 publications

Asymmetric List Sizes in Bipartite Graphs

Asymmetric List Sizes in Bipartite Graphs

Single-conflict colorings of degenerate graphs

Adaptable and conflict colouring multigraphs with no cycles of length three or four

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