A graph G is called g-perfect if, for any induced subgraph H of G, the game chromatic number of H equals the clique number of H. A graph G is called g-col-perfect if, for any induced subgraph H of G, the game coloring number of H equals the clique number of H. In this paper we characterize the classes of g-perfect resp. g-col-perfect graphs by a set of forbidden induced subgraphs. Moreover, we ...

We present new algorithms for k-coloring and minimum (x(G)-) coloring random and semi-random kcolorable graphs in polynomial expected time. The random graphs are drawn from the G(n,p, k) model and the semi-random graphs are drawn from the G s ~ ( n , p , k) model. In both models, an adversary initially splits the n vertices into IC color classes, each of size @(n). Then the edges between vertic...

A subset S ⊆ V (G) is an independent set of G if no two vertices of S are adjacent in G. A subset Q ⊆ V (G) is a 2-dominating set of G if each vertex from V (G)\Q has at least two neighbours in Q. We define new kind of kernels in graphs. Using existing concepts of an independent set and a 2-dominating set, we define in the natural way the concept of 2-dominating kernels in graphs. A subset J ⊂ ...

We recall several known results about minimally 2-connected graphs, and show that they all follow from a decomposition theorem. Starting from an analogy with critically 2-connected graphs, we give structural characterizations of the classes of graphs that do not contain as a subgraph and as an induced subgraph, a cycle with a node that has at least two neighbors on the cycle. From these charact...

A majority coloring of a directed graph is vertex-coloring in which every vertex has the same color as at most half its out-neighbors. Kreutzer, Oum, Seymour, van der Zypen and Wood proved that digraph $4$-coloring conjectured admits $3$-coloring. They observed Local Lemma implies conjecture for digraphs large enough minimum out-degree if, crucially, maximum in-degree bounded by a(n exponential...

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...

A simple graph-product type construction shows that for all natural numbers r ≥ q, there exists an edge-coloring of the complete graph on 2 vertices using r colors where the graph consisting of the union of arbitrary q color classes has chromatic number 2. We show that for each fixed natural number q, if there exists an edge-coloring of the complete graph on n vertices using r colors where the ...

We show complexity results for some generalizations of the graph coloring problem on two classes of perfect graphs, namely clique trees and unit interval graphs. We deal with the μ-coloring problem (upper bounds for the color on each vertex), the precoloring extension problem (a subset of vertices colored beforehand), and a problem generalizing both of them, the (γ, μ)-coloring problem (lower a...

A b-coloring of a graph G by k colors is a proper k-coloring of G such that in each color class there exists a vertex having neighbors in all the other k− 1 color classes. The b-chromatic number of a graph G, denoted by φ(G), is the maximum k for which G has a b-coloring by k colors. It is obvious that χ(G) ≤ φ(G). A graph G is b-continuous if for every k between χ(G) and φ(G) there is a b-colo...

An acyclic edge coloring of a graph is a proper edge coloring without bichromatic cycles. In 1978, it was conjectured that ∆(G) + 2 colors suffice for an acyclic edge coloring of every graph G [6]. The conjecture has been verified for several classes of graphs, however, the best known upper bound for as special class as planar graphs are, is ∆+12 [2]. In this paper, we study simple planar graph...