A 2-distance (2Δ+7)-coloring of planar graphs

2024/03/18 by Zakir Deniz, Deniz, Zakir
Computer Science · #Advanced Graph Theory Research #Combinatorics (math.CO) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2403.12302

openalex publication_date 2024/03/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A vertex coloring of a graph G is called a 2-distance coloring if any two vertices at a distance at most 2 from each other receive different colors. Recently, Bousquet et al. (Discrete Mathematics, 346(4), 113288, 2023) proved that 2Δ+7 colors are sufficient for the 2-distance coloring of planar graphs with maximum degree Δ≥ 9. In this paper, we strengthen their result by removing the maximum degree constraint and show that all planar graphs admit a 2-distance (2Δ+7)-coloring. This particularly improves the result of Van den Heuvel and McGuinness (Journal of Graph Theory, 42(2), 110-124, 2003).

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