List Coloring with a Bounded Palette

Bonamy, Marthe; Kang, Ross J.

doi:10.1002/jgt.22013

Cited by 5 publications

(16 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this section, we investigate general necessary conditions for (k A , k B )choosability via the complete bipartite graphs. Inspired in part by Theorem 1 and related work of Bonamy and the third author [4], this leads naturally to the study of an extremal set theoretic parameter.…”

Section: The Complete Case and Steiner Systemsmentioning

confidence: 99%

Asymmetric List Sizes in Bipartite Graphs

2021

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: The Complete Case and Steiner Systemsmentioning

confidence: 99%

Asymmetric List Sizes in Bipartite Graphs

2021

Self Cite

View full text Add to dashboard Cite

show abstract

“…This means that n is even by Lemma 31. We will derive a contradiction in the following cases: (1) n − k t = 4l for some l ∈ N, and (2) n − k t = 4l − 2 for some l ∈ N. For case (1), let G ′ = G − {v i : k t + 1 ≤ i ≤ n}. Let L ′ be the (2, 3)-assignment for G ′ obtained by restricting the domain of L to V (G ′ ).…”

Section: A Appendixmentioning

confidence: 98%

“…In 2015, it was subsequently demonstrated that this constant C must also satisfy C = Ω(4 k / √ k) as k → ∞ (see [1]). Importantly, results like this show that understanding list coloring with a bounded palette can provide us with information about list coloring in general.…”

Section: List Coloring With a Bounded Palettementioning

confidence: 99%

“…(1) k q − k q−1 − 1 = 4l for some l ∈ N, or (2) k q − k q−1 − 1 = 4l − 2 for some l ∈ N. We will derive a contradiction in each case. For case (1), let G ′ be the graph obtained from G by deleting the vertices in {v i : k q−1 + 1 ≤ i ≤ k q − 1} and then connecting v k q−1 and v kq with an edge. Let L ′ be the (2, 3)-assignment for G ′ obtained by restricting the domain of L to V (G ′ ).…”

Section: Theorem 13mentioning

confidence: 99%

“…Finally, we have three cases for L(u 1 ): (1) L(u 1 ) = {1, 2}, (2) L(u 1 ) = {1, 3}, or (3) L(u 1 ) = {2, 3}. For case (1), we can find a proportional L-coloring of G by coloring…”

Section: A Appendixmentioning

confidence: 99%

See 2 more Smart Citations

Proportional 2-Choosability with a Bounded Palette

Mudrock¹,

Piechota²,

Shin³

et al. 2019

Preprint

View full text Add to dashboard Cite

Proportional choosability is a list coloring analogue of equitable coloring. Specifically, a k-assignment L for a graph G specifies a list L(v) of k available colors to each v ∈ V (G). An L-coloring assigns a color to each vertex v from its listMotivated by earlier work, we initiate the study of proportional choosability with a bounded palette by studying proportional 2-choosability with a bounded palette. In particular, when ℓ ≥ 2, a graph G is said to be proportionallyWe show a graph is proportionally (2, 2)-choosable if and only if it is equitably 2-colorable. As ℓ gets larger, the set of propotionally (2, ℓ)-choosable graphs gets smaller. We show that whenever ℓ ≥ 5 a graph is proportionally (2, ℓ)-choosable if and only if it is proportionally 2-choosable. We also completely characterize the connected proportionally (2, ℓ)-choosable graphs when ℓ = 3, 4.

show abstract

Exact and Parameterized Algorithms for Choosability

Bliznets,

Nederlof

2024

Lecture Notes in Computer Science

View full text Add to dashboard Cite

List Coloring with a Bounded Palette

Cited by 5 publications

References 17 publications

Asymmetric List Sizes in Bipartite Graphs

Asymmetric List Sizes in Bipartite Graphs

Proportional 2-Choosability with a Bounded Palette

Exact and Parameterized Algorithms for Choosability

Contact Info

Product

Resources

About