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Polynomial bounds for centered colorings on proper minor-closed graph classes
Journal of Combinatorial Theory Series B ( IF 1.2 ) Pub Date : 2021-06-23 , DOI: 10.1016/j.jctb.2021.06.002
Michał Pilipczuk , Sebastian Siebertz

For pN, a coloring λ of the vertices of a graph G is p-centered if for every connected subgraph H of G, either H receives more than p colors under λ or there is a color that appears exactly once in H. Centered colorings play an important role in the theory of sparse graph classes introduced by Nešetřil and Ossona de Mendez [31], [32], as they structurally characterize classes of bounded expansion — one of the key sparsity notions in this theory. More precisely, a class of graphs C has bounded expansion if and only if there is a function f:NN such that every graph GC for every pN admits a p-centered coloring with at most f(p) colors. Unfortunately, known proofs for the existence of such colorings yield large upper bounds on the function f governing the number of colors needed, even for as simple classes as planar graphs. In this paper, we prove that every Kt-minor-free graph admits a p-centered coloring with O(pg(t)) colors for some function g. In the special case that the graph is embeddable in a fixed surface Σ we show that it admits a p-centered coloring with O(p19) colors, with the degree of the polynomial independent of the genus of Σ. This provides the first polynomial upper bounds on the number of colors needed in p-centered colorings of graphs drawn from proper minor-closed classes, which answers an open problem posed by Dvořák [1]. As an algorithmic application, we use our main result to prove that if C is a fixed proper minor-closed class of graphs, then given graphs H and G, on p and n vertices, respectively, where GC, it can be decided whether H is a subgraph of G in time 2O(plogp)nO(1) and space nO(1).



中文翻译:

适当次要闭合图类上居中着色的多项式边界

为了 N,着色λ的曲线图的顶点的ģ对中心,如果为每一个连通子ħģ,无论是ħ接收到多于p下的色彩,λ或存在,在恰好出现一次的彩色ħ。中心着色在 Nešetřil 和 Ossona de Mendez [31]、[32] 引入的稀疏图类理论中发挥着重要作用,因为它们在结构上表征了有界扩展的类- 该理论中的关键稀疏性概念之一。更准确地说,一类图C 有界展开当且仅当存在函数 FNN 使得每个图 GC 对于每个 N承认p中心着色最多F()颜色。不幸的是,这种着色存在的已知证据在控制所需颜色数量的函数f上产生了很大的上限,即使对于像平面图这样简单的类也是如此。在本文中,我们证明了每-minor-free 图承认p中心着色(G())某些函数g 的颜色。在图可嵌入固定表面 Σ 的特殊情况下,我们证明它允许p中心着色(19)颜色,多项式的次数与 Σ 的属无关。这提供了从适当的次闭类绘制的图的p中心着色所需颜色数量的第一个多项式上限,这回答了 Dvořák [1] 提出的一个开放问题。作为一个算法应用,我们使用我们的主要结果来证明如果C是固定的适当次闭类图,然后分别在pn顶点上给定图HG,其中GC中,可以决定是否ħ是的一个子图ģ在时间2(日志)n(1) 和空间 n(1).

更新日期:2021-06-23
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