A b-coloring is a coloring of the vertices of a graph such that each color class contains a vertex that has a neighbor in all other color classes. The b-chromatic number of a graph G is the largest integer k such that G admits a b-coloring with k colors. A graph is b-perfect if the b-chromatic number is equal to the chromatic number for every induced subgraph H of G. A graph is minimally b-impe...

An orthogonal coloring of a graph G is a pair {c1, c2} of proper colorings of G, having the property that if two vertices are colored with the same color in c1, then they must have distinct colors in c2. The notion of orthogonal colorings is strongly related to the notion of orthogonal Latin squares. The orthogonal chromatic number of G, denoted by Oχ(G), is the minimum possible number of color...

We construct perfect 2-colorings of the 12-hypercube that attain our recent bound on the dimension of arbitrary correlation immune functions. We prove that such colorings with parameters (x, 12 − x, 4 + x, 8 − x) exist if x = 0, 2, 3 and do not exist if x = 1. This is a translation into English of the original paper by D. G. Fon-Der-Flaass, “Perfect colorings of the 12-cube that attain the boun...

Certain subgraphs of a given graph G restrict the minimum number χ(G) of colors that can be assigned to the vertices of G such that the endpoints of all edges receive distinct colors. Some of such subgraphs are related to the celebrated Strong Perfect Graph Theorem, as it implies that every graph G contains a clique of size χ(G), or an odd hole or an odd anti-hole as an induced subgraph. In thi...

A graph is Berge if no induced subgraph of it is an odd cycle of length at least five or the complement of one. In joint work with Robertson, Seymour, and Thomas we recently proved the Strong Perfect Graph Theorem, which was a conjecture about the chromatic number of Berge graphs. The proof consisted of showing that every Berge graph either belongs to one of a few basic classes, or admits one o...

We study the complexity of several of the classical graph decision problems in the setting of bounded cutwidth and how imposing planarity affects the complexity. We show that for 2-coloring, for bipartite perfect matching, and for several variants of disjoint paths the straightforward NC1 upper bound may be improved to AC0[2], ACC0, and AC0 respectively for bounded planar cutwidth graphs. We ob...

The theory of perfect graphs relates the concept of graph colorings to the concept of cliques. In this paper, we introduce the concept of a perfect graph as well as a number of graph classes that are always perfect. We next introduce both theWeak Perfect Graph Theorem and the Strong Perfect Graph Theorem and provide a proof of the Weak Perfect Graph Theorem. We also demonstrate an application o...

Let us improve this bound. Assume that G is a connected graph and T is its spanning tree rooted at r. Let us consider an ordering of V (G) in which each vertex v appears after its children in T . Now, for v 6= r we have |N(vi) ∩ {v1, . . . , vi−1}| ≤ deg v − 1, so c(vi) ≤ deg vi for vi 6= r. Unfortunately, the greedy may still need to use ∆(G) + 1 colors if deg r = ∆(G) and each child of r happ...