Improved square coloring of planar graphs

Bousquet, Nicolás; Deschamps, Quentin; Meyer, Lucas de; Pierron, Théo

doi:10.1016/j.disc.2022.113288

Search citation statements

Order By: Relevance

Paper Sections

Select...

Agnarsson and Halldórsson1

Citation Types

Supporting

Mentioning

Contrasting

Year Published

2023

2024

Publication Types

Select...

Article4

Relationship

Self Cite0

Independent4

Authors

Journals

Cited by 4 publications

(1 citation statement)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Conjectured bound [22] Thomassen [20] Zhu and Bu [24] Bousquet et al [7] Wong [23] Madaras and Marcinova [16] Borodin et al [6] Theorem 1 Theorem 1. Every planar graph G with maximum degree ∆ ≥ 6 satisfies χ 2 (G) ≤ 3∆ + 4.…”

Section: Agnarsson and Halldórssonmentioning

confidence: 99%

Coloring squares of planar graphs with small maximum degree

Mateusz¹,

Rzążewski²,

Szymon³

2024

Discuss. Math. Graph Theory

View full text Add to dashboard Cite

For a graph G, by χ 2 (G) we denote the minimum integer k, such that there is a k-coloring of the vertices of G in which vertices at distance at most 2 receive distinct colors. Equivalently, χ 2 (G) is the chromatic number of the square of G. In 1977 Wegner conjectured that if G is planar and has maximum degree ∆, then χ 2 (G) ≤ 7 if ∆ ≤ 3, χ 2 (G) ≤ ∆ + 5 if 4 ≤ ∆ ≤ 7, and 3∆/2 + 1 if ∆ ≥ 8. Despite extensive work, the known upper bounds are quite far from the conjectured ones, especially for small values of ∆. In this work we show that for every planar graph G with maximum degree ∆ it holds that χ 2 (G) ≤ 3∆ + 4. This result provides the best known upper bound for 6 ≤ ∆ ≤ 14.

show abstract