Counterexamples to a Conjecture of Harris on Hall Ratio

Blumenthal, Adam; Lidický, Bernard; Martin, Ryan; Norin, Sergey; Pfender, Florian; Volec, Jan

doi:10.1137/18m1229420

Cited by 3 publications

(2 citation statements)

References 11 publications

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“…In the past 30 years, the Hall ratio has received much attention due to its connection with both list and fractional coloring (e.g. see [4,9,12,7]). Proof.…”

Section: Maximizing the Number Of Vertex Requests Satisfiedmentioning

confidence: 99%

Flexible list colorings: Maximizing the number of requests satisfied

Kaul¹,

Mathew²,

Mudrock³

et al. 2022

Preprint

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Flexible list coloring was introduced by Dvořák, Norin, and Postle in 2019. Suppose 0 ≤ ǫ ≤ 1, G is a graph, L is a list assignment for G, and r is a function with non-empty domainWe improve earlier results in different ways, often obtaining larger values of ǫ.In pursuit of the largest ǫ for which a graph is (k, ǫ)-flexible, we observe that a graph G is not (k, ǫ)-flexible for any k if and only if ǫ > 1/ρ(G), where ρ(G) is the Hall ratio of G, and we initiate the study of the list flexibility number of a graph G, which is the smallest k such that G is (k, 1/ρ(G))-flexible. We study relationships and connections between the list flexibility number, list chromatic number, list packing number, and degeneracy of a graph.

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“…In the past 30 years, the Hall ratio has received much attention due to its connection with both list and fractional coloring (e.g. see [4,9,12,7]). Proof.…”

Section: Maximizing the Number Of Vertex Requests Satisfiedmentioning

confidence: 99%

Flexible list colorings: Maximizing the number of requests satisfied

Kaul¹,

Mathew²,

Mudrock³

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

“…The Hall ratio was first studied in 1990 by Hilton and Johnson Jr. [17] under the name Hall-condition number in the context of list coloring. In the past 30 years, the Hall ratio has received much attention due to its connection with both list and fractional coloring (e.g., see [4,7,10,13]).…”

Section: Maximizing the Number Of Vertex Requests Satisfiedmentioning

confidence: 99%

Flexible list colorings: Maximizing the number of requests satisfied

Kaul,

Mathew,

Mudrock

et al. 2024

Journal of Graph Theory

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Flexible list coloring was introduced by Dvořák, Norin, and Postle in 2019. Suppose , is a graph, is a list assignment for , and is a function with nonempty domain such that for each ( is called a request of ). The triple is ‐satisfiable if there exists a proper ‐coloring of such that for at least vertices in . We say is ‐flexible if is ‐satisfiable whenever is a ‐assignment for and is a request of . It was shown by Dvořák et al. that if is prime, is a ‐degenerate graph, and is a request for with domain of size 1, then is 1‐satisfiable whenever is a ‐assignment. In this paper, we extend this result to all for bipartite ‐degenerate graphs.The literature on flexible list coloring tends to focus on showing that for a fixed graph and there exists an such that is ‐flexible, but it is natural to try to find the largest possible for which is ‐flexible. In this vein, we improve a result of Dvořák et al., by showing ‐degenerate graphs are ‐flexible. In pursuit of the largest for which a graph is ‐flexible, we observe that a graph is not ‐flexible for any if and only if , where is the Hall ratio of , and we initiate the study of the list flexibility number of a graph , which is the smallest such that is ‐flexible. We study relationships and connections between the list flexibility number, list chromatic number, list packing number, and degeneracy of a graph.

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1-subdivisions, fractional chromatic number and Hall ratio

Dvořák

Mendez²,

2018

Preprint

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The Hall ratio of a graph G is the maximum of |V (H)|/α(H) over all subgraphs H of G. Clearly, the Hall ratio of a graph is a lower bound for the fractional chromatic number. It has been asked whether conversely, the fractional chromatic number is upper bounded by a function of the Hall ratio. We answer this question in negative, by showing two results of independent interest regarding 1-subdivisions (the 1-subdivision of a graph is obtained by subdividing each edge exactly once).

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Counterexamples to a Conjecture of Harris on Hall Ratio

Cited by 3 publications

References 11 publications

Flexible list colorings: Maximizing the number of requests satisfied

Flexible list colorings: Maximizing the number of requests satisfied

Flexible list colorings: Maximizing the number of requests satisfied

1-subdivisions, fractional chromatic number and Hall ratio

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