A short note on conflict‐free coloring on closed neighborhoods of bounded degree graphs

Bhyravarapu, Sriram; Kalyanasundaram, Subrahmanyam; Mathew, Rogers

doi:10.1002/jgt.22670

Cited by 4 publications

(2 citation statements)

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“…Note that this result gives a logarithmic (in

normalΔ

) lower bound on the conflict‐free chromatic number for a specific family of graphs, unlike other lower bounds mentioned, based on probabilistic constructions. We remark that an open neighborhood analog of the following theorem has been proved in an independent work by Bhyravarapu, Kalyanasundaram, and Mathew [3].…”

Section: Introductionmentioning

confidence: 84%

“…In 2009 Pach and Tardos showed that the answer is of order at most

\ln^{2 + ϵ} normalΔ

and at least

\ln normalΔ

[10]. Both bounds have been improved to

normalΘ (\ln^{2} Δ)

; Glebov, Szabó, and Tardos showed that certain random graphs on

n

vertices require at least

normalΩ (\ln^{2} n)

colors [5], while Bhyravarapu, Kalyanasundaram, and Mathew gave a randomized procedure that constructs conflict‐free colorings using

O (\ln^{2} Δ)

colors [2]. Constants hidden in the

normalΩ, O

‐notations are different, as one might expect from probabilistic proofs, but if we focus only on the order of magnitude—the question is completely answered and the mentioned results can be thought of as a conflict‐free analog of Brooks’ theorem.…”

Section: Introductionmentioning

confidence: 99%

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