Bounded Max-Colorings of Graphs

In a bounded max-coloring of a vertex/edge weighted graph, each color class is of cardinality at most $b$ and of weight equal to the weight of the heaviest vertex/edge in this class. The bounded max-vertex/edge-coloring problems ask for such a coloring minimizing the sum of all color classes' weights. In this paper we present complexity results and approximation algorithms for those problems on general graphs, bipartite graphs and trees. We first show that both problems are polynomial for trees, when the number of colors is fixed, and $H_b$ approximable for general graphs, when the bound $b$ is fixed. For the bounded max-vertex-coloring problem, we show a 17/11-approximation algorithm for bipartite graphs, a PTAS for trees as well as for bipartite graphs when $b$ is fixed. For unit weights, we show that the known 4/3 lower bound for bipartite graphs is tight by providing a simple 4/3 approximation algorithm. For the bounded max-edge-coloring problem, we prove approximation factors of $3-2/\sqrt{2b}$, for general graphs, $\min\{e, 3-2/\sqrt{b}\}$, for bipartite graphs, and 2, for trees. Furthermore, we show that this problem is NP-complete even for trees. This is the first complexity result for max-coloring problems on trees.Comment: 13 pages, 5 figure

Spatial Mixing of Coloring Random Graphs

We study the strong spatial mixing (decay of correlation) property of proper $q$-colorings of random graph $G(n, d/n)$ with a fixed $d$. The strong spatial mixing of coloring and related models have been extensively studied on graphs with bounded maximum degree. However, for typical classes of graphs with bounded average degree, such as $G(n, d/n)$, an easy counterexample shows that colorings do not exhibit strong spatial mixing with high probability. Nevertheless, we show that for $q\ge\alpha d+\beta$ with $\alpha>2$ and sufficiently large $\beta=O(1)$, with high probability proper $q$-colorings of random graph $G(n, d/n)$ exhibit strong spatial mixing with respect to an arbitrarily fixed vertex. This is the first strong spatial mixing result for colorings of graphs with unbounded maximum degree. Our analysis of strong spatial mixing establishes a block-wise correlation decay instead of the standard point-wise decay, which may be of interest by itself, especially for graphs with unbounded degree

Grad and classes with bounded expansion I. decompositions

We introduce classes of graphs with bounded expansion as a generalization of both proper minor closed classes and degree bounded classes. Such classes are based on a new invariant, the greatest reduced average density (grad) of G with rank r, grad r(G). For these classes we prove the existence of several partition results such as the existence of low tree-width and low tree-depth colorings. This generalizes and simplifies several earlier results (obtained for minor closed classes)

Three-coloring triangle-free graphs on surfaces V. Coloring planar graphs with distant anomalies

We settle a problem of Havel by showing that there exists an absolute constant d such that if G is a planar graph in which every two distinct triangles are at distance at least d, then G is 3-colorable. In fact, we prove a more general theorem. Let G be a planar graph, and let H be a set of connected subgraphs of G, each of bounded size, such that every two distinct members of H are at least a specified distance apart and all triangles of G are contained in \bigcup{H}. We give a sufficient condition for the existence of a 3-coloring phi of G such that for every B\in H, the restriction of phi to B is constrained in a specified way.Comment: 26 pages, no figures. Updated presentatio

Product Dimension of Forests and Bounded Treewidth Graphs

The product dimension of a graph G is defined as the minimum natural number l such that G is an induced subgraph of a direct product of l complete graphs. In this paper we study the product dimension of forests, bounded treewidth graphs and k-degenerate graphs. We show that every forest on n vertices has a product dimension at most 1.441logn+3. This improves the best known upper bound of 3logn for the same due to Poljak and Pultr. The technique used in arriving at the above bound is extended and combined with a result on existence of orthogonal Latin squares to show that every graph on n vertices with a treewidth at most t has a product dimension at most (t+2)(logn+1). We also show that every k-degenerate graph on n vertices has a product dimension at most \ceil{8.317klogn}+1. This improves the upper bound of 32klogn for the same by Eaton and Rodl.Comment: 12 pages, 3 figure