Occupancy fraction, fractional colouring, and triangle fraction

Davies, Ewan; Verclos, Rémi de Joannis de; Kang, Ross J.; Pirot, François

doi:10.1002/jgt.22671

Cited by 4 publications

(2 citation statements)

References 15 publications

(35 reference statements)

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“…We need a lower bound on s α G pλq for certain range of λ, due to Davies et al [11]. The lower bound is written in terms of the Lambert W function: for z ą 0, W pzq is the unique positive real satisfying W pzqe W pzq " z.…”

Section: Graph Theoretic Toolsmentioning

confidence: 99%

Exponential decay of intersection volume with applications on list-decodability and Gilbert-Varshamov type bound

Kim¹,

Liu²,

Tran³

2021

Preprint

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We give some natural sufficient conditions for balls in a metric space to have small intersection. Roughly speaking, this happens when the metric space is (i) expanding and (ii) well-spread, and (iii) a certain random variable on the boundary of a ball has a small tail. As applications, we show that the volume of intersection of balls in Hamming, Johnson spaces and symmetric groups decay exponentially as their centers drift apart. To verify condition (iii), we prove some large deviation inequalities 'on a slice' for functions with Lipschitz conditions.We then use these estimates on intersection volumes to• obtain a sharp lower bound on list-decodability of random q-ary codes, confirming a conjecture of Li and Wootters; and• improve the classical bound of Levenshtein from 1971 on constant weight codes by a factor linear in dimension, resolving a problem raised by Jiang and Vardy.Our probabilistic point of view also offers a unified framework to obtain improvements on other Gilbert-Varshamov type bounds, giving conceptually simple and calculation-free proofs for q-ary codes, permutation codes, and spherical codes. Another consequence is a counting result on the number of codes, showing ampleness of large codes.

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Section: Graph Theoretic Toolsmentioning

confidence: 99%

Exponential decay of intersection volume with applications on list-decodability and Gilbert-Varshamov type bound

Kim¹,

Liu²,

Tran³

2021

Preprint

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“…This too has been refined recently [10] using elementary properties of the hard-core model (cf. also [8] and [9]) as follows: any graph G with local density at most 1/ f , where f = f (∆(G)), f → ∞ as ∆(G) → ∞, and f ≤ ∆(G) 2 + 1, has chromatic number satisfying χ(G) ≤ (1 + o(1))∆(G)/ log √ f . Note that this statement includes the triangle-free one as a special case with f = ∆(G) 2 + 1.…”

Section: Introductionmentioning

confidence: 99%

An improved procedure for colouring graphs of bounded local density

Hurley¹,

Verclos²,

Kang³

2022

AiC

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We develop an improved bound for the chromatic number of graphs of maximum degree ∆ under the assumption that the number of edges spanning any neighbourhood is at most (1 − σ ) ∆ 2 for some fixed 0 < σ < 1. The leading term in the reduction of colours achieved through this bound is best possible as σ → 0. As two consequences, we advance the state of the art in two longstanding and well-studied graph colouring conjectures, the Erdős-Nešetřil conjecture and Reed's conjecture. We prove that the strong chromatic index is at most 1.772∆ 2 for any graph G with sufficiently large maximum degree ∆. We prove that the chromatic number is at most 0.881(∆ + 1) + 0.119ω for any graph G with clique number ω and sufficiently large maximum degree ∆. Additionally, we show how our methods can be adapted under the additional assumption that the codegree is at most (1 − σ )∆, and establish what may be considered first progress towards a conjecture of Vu.

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