Cooperative colorings of trees and of bipartite graphs

Ron Aharoni; Eli Berger; Maria Chudnovsky; Frédéric Havet; Zilin Jiang

doi:10.37236/8111

Cooperative colorings of trees and of bipartite graphs

Ron Aharoni, Eli Berger, Maria Chudnovsky, Frédéric Havet, Zilin Jiang

Research output: Contribution to journal › Article › peer-review

4 Scopus citations

Abstract

Given a system (G₁, …, G_m) of graphs on the same vertex set V, a cooperative coloring is a choice of vertex sets I₁, …, I_m, such that I_j is independent in G_j and (formula presented)_m j=1^Ij = V . For a class G of graphs, let m_G (d) be the minimal m such that every m graphs from G with maximum degree d have a cooperative coloring. We prove that Ω(log log d) ≤ m_T (d) ≤ O(log d) and Ω(log d) ≤ m_B(d) ≤ O(d/ log d), where T is the class of trees and B is the class of bipartite graphs.

Original language	English (US)
Article number	P1.41
Journal	Electronic Journal of Combinatorics
Volume	27
Issue number	1
DOIs	https://doi.org/10.37236/8111
State	Published - 2020
Externally published	Yes

ASJC Scopus subject areas

Theoretical Computer Science
Geometry and Topology
Discrete Mathematics and Combinatorics
Computational Theory and Mathematics
Applied Mathematics

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10.37236/8111

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@article{aba2a1be14d64808a725c9959c45e3de,

title = "Cooperative colorings of trees and of bipartite graphs",

abstract = "Given a system (G1, …, Gm) of graphs on the same vertex set V, a cooperative coloring is a choice of vertex sets I1, …, Im, such that Ij is independent in Gj and (formula presented)m j=1Ij = V . For a class G of graphs, let mG (d) be the minimal m such that every m graphs from G with maximum degree d have a cooperative coloring. We prove that Ω(log log d) ≤ mT (d) ≤ O(log d) and Ω(log d) ≤ mB(d) ≤ O(d/ log d), where T is the class of trees and B is the class of bipartite graphs.",

author = "Ron Aharoni and Eli Berger and Maria Chudnovsky and Fr{\'e}d{\'e}ric Havet and Zilin Jiang",

note = "Funding Information: Supported in part by the United States–Israel Binational Science Foundation (BSF) grant no. 2006099, the Israel Science Foundation (ISF) grant no. 2023464 and the Discount Bank Chair at Technion. This paper is part of a project that has received funding from the European Union{\textquoteright}s Horizon 2020 research and innovation programme, under the Marie Sklodowska-Curie grant agreement no. 823748.†Supported in part by BSF grant no. 2006099 and ISF grant no. 2023464.‡Supported in part by BSF grant no. 2006099, NSF grant DMS-1550991 and US Army Research Office Grant W911NF-16-1-0404.§The work was done when Z. Jiang was a postdoctoral fellow at Technion – Israel Institute of Technology, and was supported in part by ISF grant nos. 409/16, 936/16. Funding Information: ∗Supported in part by the United States–Israel Binational Science Foundation (BSF) grant no. 2006099, the Israel Science Foundation (ISF) grant no. 2023464 and the Discount Bank Chair at Technion. This paper is part of a project that has received funding from the European Union{\textquoteright}s Horizon 2020 research and innovation programme, under the Marie Sk lodowska-Curie grant agreement no. 823748. †Supported in part by BSF grant no. 2006099 and ISF grant no. 2023464. ‡Supported in part by BSF grant no. 2006099, NSF grant DMS-1550991 and US Army Research Office Grant W911NF-16-1-0404. §The work was done when Z. Jiang was a postdoctoral fellow at Technion – Israel Institute of Technology, and was supported in part by ISF grant nos. 409/16, 936/16. Publisher Copyright: {\textcopyright} The authors. Released under the CC BY-ND license (International 4.0).",

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AU - Berger, Eli

AU - Chudnovsky, Maria

AU - Havet, Frédéric

AU - Jiang, Zilin

N1 - Funding Information: Supported in part by the United States–Israel Binational Science Foundation (BSF) grant no. 2006099, the Israel Science Foundation (ISF) grant no. 2023464 and the Discount Bank Chair at Technion. This paper is part of a project that has received funding from the European Union’s Horizon 2020 research and innovation programme, under the Marie Sklodowska-Curie grant agreement no. 823748.†Supported in part by BSF grant no. 2006099 and ISF grant no. 2023464.‡Supported in part by BSF grant no. 2006099, NSF grant DMS-1550991 and US Army Research Office Grant W911NF-16-1-0404.§The work was done when Z. Jiang was a postdoctoral fellow at Technion – Israel Institute of Technology, and was supported in part by ISF grant nos. 409/16, 936/16. Funding Information: ∗Supported in part by the United States–Israel Binational Science Foundation (BSF) grant no. 2006099, the Israel Science Foundation (ISF) grant no. 2023464 and the Discount Bank Chair at Technion. This paper is part of a project that has received funding from the European Union’s Horizon 2020 research and innovation programme, under the Marie Sk lodowska-Curie grant agreement no. 823748. †Supported in part by BSF grant no. 2006099 and ISF grant no. 2023464. ‡Supported in part by BSF grant no. 2006099, NSF grant DMS-1550991 and US Army Research Office Grant W911NF-16-1-0404. §The work was done when Z. Jiang was a postdoctoral fellow at Technion – Israel Institute of Technology, and was supported in part by ISF grant nos. 409/16, 936/16. Publisher Copyright: © The authors. Released under the CC BY-ND license (International 4.0).

PY - 2020

Y1 - 2020

N2 - Given a system (G1, …, Gm) of graphs on the same vertex set V, a cooperative coloring is a choice of vertex sets I1, …, Im, such that Ij is independent in Gj and (formula presented)m j=1Ij = V . For a class G of graphs, let mG (d) be the minimal m such that every m graphs from G with maximum degree d have a cooperative coloring. We prove that Ω(log log d) ≤ mT (d) ≤ O(log d) and Ω(log d) ≤ mB(d) ≤ O(d/ log d), where T is the class of trees and B is the class of bipartite graphs.

AB - Given a system (G1, …, Gm) of graphs on the same vertex set V, a cooperative coloring is a choice of vertex sets I1, …, Im, such that Ij is independent in Gj and (formula presented)m j=1Ij = V . For a class G of graphs, let mG (d) be the minimal m such that every m graphs from G with maximum degree d have a cooperative coloring. We prove that Ω(log log d) ≤ mT (d) ≤ O(log d) and Ω(log d) ≤ mB(d) ≤ O(d/ log d), where T is the class of trees and B is the class of bipartite graphs.

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ER -

Cooperative colorings of trees and of bipartite graphs

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