Improved bounds for centered colorings

Abstract

A vertex coloring φ of a graph G is p-centered if for every connected subgraph H of G either φ uses more than p colors on H or there is a color that appears exactly once on H. Centered colorings form one of the families of parameters that allow to capture notions of sparsity of graphs: A class of graphs has bounded expansion if and only if there is a function f such that for every p 1, every graph in the class admits a p-centered coloring using at most f (p) colors.In this paper, we give upper bounds for the maximum number of colors needed in a p-centered coloring of graphs from several widely studied graph classes. We show that:(1) planar graphs admit p-centered colorings with O(p 3 log p) colors where the previous bound was O(p 19 ); (2) bounded degree graphs admit p-centered colorings with O(p) colors while it was conjectured that they may require exponential number of colors in p;(3) graphs avoiding a fixed graph as a topological minor admit p-centered colorings with a polynomial in p number of colors. All these upper bounds imply polynomial algorithms for computing the colorings. Prior to this work there were no non-trivial lower bounds known. We show that: (4) there are graphs of treewidth t that require p+t t colors in any p-centered coloring and this bound matches the upper bound; (5) there are planar graphs that require Ω(p 2 log p) colors in any p-centered coloring. We also give asymptotically tight bounds for outerplanar graphs and planar graphs of treewidth 3. We prove our results with various proof techniques. The upper bound for planar graphs involves an application of a recent structure theorem while the upper bound for bounded degree graphs comes from the entropy compression method. We lift the result for bounded degree graphs to graphs avoiding a fixed topological minor using the Grohe-Marx structure theorem.

Improved Bounds for Centered Colorings

“…For every graph H, every graph excluding H as a topological minor has a tree-decomposition with bounded adhesion such that every torso either has bounded degree with the exception of a bounded number of vertices, or excludes a fixed graph as a minor. Furthermore, such a decomposition for a graph G can be computed in time f (H) 1) for some computable function f . We proceed with all the necessary definitions.…”

“…A coloring procedure which follows the proof of Theorem 9 more closely can be used to show that graphs of Euler genus g actually admit p-centered colorings with O(gp + p 3 log p) colors. The proof for this slightly improved bound has been detailed in the conference version of this paper [1]. There it is also indicated how the proofs can be turned into quadratic time algorithms for colorings which respect the given bounds for the number of colors.…”

“…Low treedepth colorings are a central tool for designing parametrized algorithms in classes of bounded expansion. For example, Pilipczuk and Siebertz [21] showed that when C is a class of graphs avoiding a fixed minor, then it can be decided whether a given p-vertex graph H is a subgraph of a given n-vertex graph G in C in time 2 O(p log p) • n O (1) and space n O (1) . The algorithm witnessing this statement starts with a computation of a p-centered coloring of G with p O (1) colors, and for each p-tuple of colors it applies a procedure to solve this problem for graphs G of treedepth at most p. The results of our paper imply a corresponding algorithm for the case where C is a class of graphs avoiding a fixed graph as a topological minor.…”

Notes on Graph Product Structure Theory

“…We do not define "p-centred colouring" here since we do not need the definition. All we need to know is that subgraphs of H ⊠ P , where H has bounded treewidth and P is a path, have p-centred colourings with f (p) colours, where f is a polynomial function (see [10,16]). This result is easily extended to allow for apex vertices and clique sums.…”

“…• Dujmović, Eppstein, Joret, Morin, and Wood [12] use Theorem 1 to show that planar graphs can be nonrepetitively coloured with a bounded number of colours, solving a 17 year old problem of Alon, Grytczuk, Hałuszczak, and Riordan [1]. • Dębski, Felsner, Micek, and Schröder [10] use Theorem 1 to prove the best known results on p-centred colourings of planar graphs, reducing the bound from O(p 19 ) to O(p 3 log p). • Bonamy, Gavoille, and Pilipczuk [5] use Theorem 1 to give more compact graph encodings of planar graphs.…”