The list version of the Borodin-Kostochka conjecture for graphs with large maximum degree

Ilkyoo Choi; H. A. Kierstead; Landon Rabern

doi:10.1016/j.disc.2022.113300

The list version of the Borodin-Kostochka conjecture for graphs with large maximum degree

Ilkyoo Choi, H. A. Kierstead, Landon Rabern

Mathematical and Statistical Sciences, School of (SoMSS)

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

1 Scopus citations

Abstract

Brooks' Theorem states that the chromatic number χ(G) of a graph G is at most its maximum degree Δ(G) when Δ(G)≥3 and its clique number ω(G) is at most Δ(G). Vizing strengthened this by proving that the list chromatic number χ_l(G) is also at most Δ(G) for these graphs. Borodin and Kostochka conjectured that χ(G)<Δ(G) for all graphs G with Δ(G)≥9 and ω(G)<Δ(G). Reed proved their conjecture for graphs G with Δ(G) sufficiently large. Here we prove the analogous result for list coloring: χ_l(G)<Δ(G) for all graphs G with Δ(G) sufficiently large and ω(G)<Δ(G).

Original language	English (US)
Article number	113300
Journal	Discrete Mathematics
DOIs	https://doi.org/10.1016/j.disc.2022.113300
State	Accepted/In press - 2023

Keywords

Borodin-Kostochka conjecture
Bounded maximum degree
List coloring

ASJC Scopus subject areas

Theoretical Computer Science
Discrete Mathematics and Combinatorics

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10.1016/j.disc.2022.113300

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title = "The list version of the Borodin-Kostochka conjecture for graphs with large maximum degree",

abstract = "Brooks' Theorem states that the chromatic number χ(G) of a graph G is at most its maximum degree Δ(G) when Δ(G)≥3 and its clique number ω(G) is at most Δ(G). Vizing strengthened this by proving that the list chromatic number χl(G) is also at most Δ(G) for these graphs. Borodin and Kostochka conjectured that χ(G)<Δ(G) for all graphs G with Δ(G)≥9 and ω(G)<Δ(G). Reed proved their conjecture for graphs G with Δ(G) sufficiently large. Here we prove the analogous result for list coloring: χl(G)<Δ(G) for all graphs G with Δ(G) sufficiently large and ω(G)<Δ(G).",

keywords = "Borodin-Kostochka conjecture, Bounded maximum degree, List coloring",

author = "Ilkyoo Choi and Kierstead, {H. A.} and Landon Rabern",

note = "Funding Information: Supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2018R1D1A1B07043049), and also by the Hankuk University of Foreign Studies Research Fund. Publisher Copyright: {\textcopyright} 2022 Elsevier B.V.",

year = "2023",

doi = "10.1016/j.disc.2022.113300",

language = "English (US)",

journal = "Discrete Mathematics",

issn = "0012-365X",

publisher = "Elsevier",

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T1 - The list version of the Borodin-Kostochka conjecture for graphs with large maximum degree

AU - Choi, Ilkyoo

AU - Kierstead, H. A.

AU - Rabern, Landon

N1 - Funding Information: Supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2018R1D1A1B07043049), and also by the Hankuk University of Foreign Studies Research Fund. Publisher Copyright: © 2022 Elsevier B.V.

PY - 2023

Y1 - 2023

N2 - Brooks' Theorem states that the chromatic number χ(G) of a graph G is at most its maximum degree Δ(G) when Δ(G)≥3 and its clique number ω(G) is at most Δ(G). Vizing strengthened this by proving that the list chromatic number χl(G) is also at most Δ(G) for these graphs. Borodin and Kostochka conjectured that χ(G)<Δ(G) for all graphs G with Δ(G)≥9 and ω(G)<Δ(G). Reed proved their conjecture for graphs G with Δ(G) sufficiently large. Here we prove the analogous result for list coloring: χl(G)<Δ(G) for all graphs G with Δ(G) sufficiently large and ω(G)<Δ(G).

AB - Brooks' Theorem states that the chromatic number χ(G) of a graph G is at most its maximum degree Δ(G) when Δ(G)≥3 and its clique number ω(G) is at most Δ(G). Vizing strengthened this by proving that the list chromatic number χl(G) is also at most Δ(G) for these graphs. Borodin and Kostochka conjectured that χ(G)<Δ(G) for all graphs G with Δ(G)≥9 and ω(G)<Δ(G). Reed proved their conjecture for graphs G with Δ(G) sufficiently large. Here we prove the analogous result for list coloring: χl(G)<Δ(G) for all graphs G with Δ(G) sufficiently large and ω(G)<Δ(G).

KW - Borodin-Kostochka conjecture

KW - Bounded maximum degree

KW - List coloring

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JO - Discrete Mathematics

JF - Discrete Mathematics

M1 - 113300

ER -

The list version of the Borodin-Kostochka conjecture for graphs with large maximum degree

Abstract

Keywords

ASJC Scopus subject areas

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