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Two lower bounds for p-centered colorings

par Dubois, Loïc;Joret, Gwenaël ;Perarnau, Guillem;Pilipczuk, Marcin;Pitois, François
Référence Discrete mathematics and theoretical computer science, 22, 4
Publication Publié, 2020-12-01

Article révisé par les pairs

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Résumé :

Given a graph G and an integer p, a coloring f : V (G) N is p-centered if for every connected subgraph H of G, either f uses more than p colors on H or there is a color that appears exactly once in H. The notion of p-centered colorings plays a central role in the theory of sparse graphs. In this note we show two lower bounds on the number of colors required in a p-centered coloring. First, we consider monotone classes of graphs whose shallow minors have average degree bounded polynomially in the radius, or equivalently (by a result of Dvǒrák and Norin), admitting strongly sublinear separators. We construct such a class such that p-centered colorings require a number of colors super-polynomial in p. This is in contrast with a recent result of Pilipczuk and Siebertz, who established a polynomial upper bound in the special case of graphs excluding a fixed minor. Second, we consider graphs of maximum degree. Debski, Felsner, Micek, and Schröder recently proved that these graphs have p-centered colorings with O(21=pp) colors. We show that there are graphs of maximum degree that require (21=pp ln1=p) colors in any p-centered coloring, thus matching their upper bound up to a logarithmic factor.