This paper describes a new exact algorithm for the Equitable Coloring Problem, a coloring problem where the sizes of two arbitrary color classes differ in at most one unit. Based on the well known DSatur algorithm for the classic Coloring Problem, a pruning criterion arising from equity constraints is proposed and analyzed. The good performance of the algorithm is shown through computational ex...

The game domination number is a graph invariant that arises from a game, which is related to graph domination in a similar way as the game chromatic number is related to graph coloring. In this paper we show that verifying whether the game domination number of a graph is bounded by a given integer is PSPACEcomplete. This contrasts the situation of the game coloring problem whose complexity is s...

A graph G is 2-stratified if its vertex set is colored into two nonempty classes, where one class of vertices colored red and the other color class blue. Let F be a 2-stratified graph rooted at one fixed blue vertex v. The F -domination number of G is the minimum number of red vertices of G in a red-blue coloring of the vertices of G such that for every blue vertex v of G, there is a copy of F ...

In this paper, we show that the domination number D of a random graph enjoys as sharp a concentration as does its chromatic number χ. We first prove this fact for the sequence of graphs {G(n, pn}, n → ∞, where a two point concentration is obtained with high probability for pn = p (fixed) or for a sequence pn that approaches zero sufficiently slowly. We then consider the infinite graph G(Z+, p),...

A graph G is 2-stratified if its vertex set is partitioned into two classes (each of which is a stratum or a color class). We color the vertices in one color class red and the other color class blue. Let F be a 2-stratified graph with one fixed blue vertex v specified. We say that F is rooted at v. The Fdomination number of a graphG is the minimum number of red vertices ofG in a red-blue colori...

In Defective Coloring we are given a graph G and two integers χd,∆ ∗ and are asked if we can χd-color G so that the maximum degree induced by any color class is at most ∆∗. We show that this natural generalization of Coloring is much harder on several basic graph classes. In particular, we show that it is NP-hard on split graphs, even when one of the two parameters χd,∆ ∗ is set to the smallest...

The AutoGraphiX 2 system is used to compare the index of a connected graph G with a number of other graph theoretical invariants, i.e., chromatic number, maximum, minimum and average degree, diameter, radius, average distance, independence and domination numbers. In each case, best possible lower and upper bounds, in terms of the order of G, are sought for sums, differences, ratios and products...

In this paper we present upper bounds on the differences between the independence, domination and irredundance parameters of a graph. For example, using the Brooks theorem on the chromatic number, we show that for any graph G of order n with maximum degree ∆ ≥ 2 IR(G)− β(G) ≤ ⌊ ∆− 2 2∆ n ⌋ , where β(G) and IR(G) are the independence number and the upper irredundance number of a graph G, respect...

The recently introduced graph parameter tree-cut width plays a similar role with respect to immersions as the graph parameter treewidth plays with respect to minors. In this paper we provide the first algorithmic applications of tree-cut width to hard combinatorial problems. Tree-cut width is known to be lower-bounded by a function of treewidth, but it can be much larger and hence has the poten...

A graph G is 2-stratified if its vertex set is partitioned into two classes (each of which is a stratum or a color class), where the vertices in one class are colored red and those in the other class are colored blue. Let F be a 2-stratified graph rooted at some blue vertex v. An F -coloring of a graph is a red-blue coloring of the vertices of G in which every blue vertex v belongs to a copy of...