Abstract The weak r -coloring numbers wcol r ( G ) of a graph G were introduced by the first two authors as a generalization of the usual coloring number col ( G ) , and have since found interesting theoretical and algorithmic applications. This has motivated researchers to establish strong bounds on these parameters for various classes of graphs. Let G p denote the p th power of G . We show that, all integers p > 0 and Δ ≥ 3 and graphs G with Δ ( G ) ≤ Δ satisfy col ( G p ) ∈ O ( p ⋅ wcol ⌈ p ∕ 2 ⌉ ( G ) ( Δ − 1 ) ⌊ p ∕ 2 ⌋ ) ; for fixed tree width or fixed genus the ratio between this upper bound and worst case lower bounds is polynomial in p . For the square of graphs G , we also show that, if the maximum average degree 2 k − 2 mad ( G ) ≤ 2 k , then col ( G 2 ) ≤ ( 2 k − 1 ) Δ ( G ) + 2 k + 1 .

中文翻译：

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关于图幂的着色数

摘要 图 G 的弱 r 着色数 wcol r ( G ) 由前两位作者引入，作为通常着色数 col ( G ) 的推广，此后发现了有趣的理论和算法应用。这促使研究人员为各种类型的图建立这些参数的强界限。令 G p 表示 G 的 p 次幂。我们证明，所有整数 p > 0 和 Δ ≥ 3 以及具有 Δ ( G ) ≤ Δ 的图 G 满足 col ( G p ) ∈ O ( p ⋅ wcol ⌈ p ∕ 2 ⌉ ( G ) ( Δ − 1 ) ⌊ p ∕ 2 ⌋ ) ; 对于固定树宽或固定属，此上限与最坏情况下限之间的比率是 p 中的多项式。对于图 G 的平方，我们还证明，如果最大平均度 2 k − 2 mad ( G ) ≤ 2 k ，则 col ( G 2 ) ≤ ( 2 k − 1 ) Δ ( G ) + 2 k + 1 .