2-Distance (Δ + 1)-coloring of sparse graphs using the potential method

IF 0.7 3区 数学 Q2 MATHEMATICS Discrete Mathematics Pub Date : 2024-10-21 DOI:10.1016/j.disc.2024.114292
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Abstract

A 2-distance k-coloring of a graph is a proper k-coloring of the vertices where vertices at distance at most 2 cannot share the same color. We prove the existence of a 2-distance (Δ+1)-coloring for graphs with maximum average degree less than 187 and maximum degree Δ7. As a corollary, every planar graph with girth at least 9 and Δ7 admits a 2-distance (Δ+1)-coloring. The proof uses the potential method to reduce new configurations compared to classic approaches on 2-distance coloring.
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使用势能法对稀疏图进行 2-Distance (Δ + 1) 着色
图的 2-distance k-coloring 是顶点的适当 k-coloring ,其中顶点的距离最多为 2,不能共享相同的颜色。我们证明了最大平均度数小于 187 且最大度数Δ≥7 的图的 2-distance (Δ+1)-coloring 的存在。作为推论,每个周长至少为 9 且 Δ≥7 的平面图都有一个 2-distance (Δ+1)-coloring 。与经典的 2-距离着色方法相比,证明使用了势法来减少新的配置。
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来源期刊
Discrete Mathematics
Discrete Mathematics 数学-数学
CiteScore
1.50
自引率
12.50%
发文量
424
审稿时长
6 months
期刊介绍: Discrete Mathematics provides a common forum for significant research in many areas of discrete mathematics and combinatorics. Among the fields covered by Discrete Mathematics are graph and hypergraph theory, enumeration, coding theory, block designs, the combinatorics of partially ordered sets, extremal set theory, matroid theory, algebraic combinatorics, discrete geometry, matrices, and discrete probability theory. Items in the journal include research articles (Contributions or Notes, depending on length) and survey/expository articles (Perspectives). Efforts are made to process the submission of Notes (short articles) quickly. The Perspectives section features expository articles accessible to a broad audience that cast new light or present unifying points of view on well-known or insufficiently-known topics.
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