Consider families of arcs on a circle. The minimum coloring problem on arc families has been shown to be NP-hard by Garey, Johnson, Miller and Papadimitriou. It is easy to show that 2q colors are sufficient for any arc family F, where q is the size of a maximum clique in F and 3q/2 colors are necessary for some families. It has long been open problem to find a coloring algorithm which uses no m...

Gyárfás et al. and Zaker have proven that the Grundy number of a graph G satisfies Γ(G) ≥ t if and only if G contains an induced subgraph called a t-atom. The family of t-atoms has bounded order and contains a finite number of graphs. In this article, we introduce equivalents of tatoms for b-coloring and partial Grundy coloring. This concept is used to prove that determining if φ(G) ≥ t and ∂Γ(...

The domatic partition problem seeks to maximize the partitioning of the nodes of the network into disjoint dominating sets. These sets represent a series of virtual backbones for wireless sensor networks to be activated successively, resulting in more balanced energy consumption and increased network robustness. In this study, we address the domatic partition problem in random geometric graphs ...

the robust coloring problem (rcp) is a generalization of the well-known graph coloring problem where we seek for a solution that remains valid when extra edges are added. the rcp is used in scheduling of events with possible last-minute changes and study frequency assignments of the electromagnetic spectrum. this problem has been proved as np-hard and in instances larger than 30 vertices, meta-...

We consider the offline and online versions of a bin packing problem called bin packing with conflicts. Given a set of items V = {1, 2, . . . , n} with sizes s1, s2 . . . , sn ∈ [0, 1] and a conflict graph G = (V, E), the goal is to find a partition of the items into independent sets of G, where the total size of each independent set is at most one, so that the number of independent sets in the...

The chromatic index of a bipartite graph equals the maximal degree of its vertices. The straightforward way to compute the corresponding edge coloring using colors, requires O((2 n 3=2) time. We will show that a simple divide & conquer algorithm only requires O((3=2 n 3=2) time. This algorithm uses an algorithm for perfect k-matching in regular bipartite graphs as a sub-routine. We will show th...

Let T = (V,A) be a directed tree. Given a collection P of dipaths on T , we can look at the arc-intersection graph I(P, T ) whose vertex set is P and where two vertices are connected by an edge if the corresponding dipaths share a common arc. Monma and Wei, who started their study in a seminal paper on intersection graphs of paths on a tree, called them DE graphs (for directed edge path graphs)...

In the “mixed hypergraph” model, proper coloring requires that vertex subsets of one type (called C-edges) should contain two vertices of the same color, while the other type (D-edges) should not be monochromatic. Voloshin [Australas. J. Combin. 11 (1995), 25–45] introduced the concept of C-perfectness, which can be viewed as a dual kind of graph perfectness in the classical sense, and proposed...

A vertex coloring of a graph is said to be perfect with parameters (aij) k i,j=1 if for every i, j ∈ {1, ..., k} every vertex of color i is adjacent with exactly aij vertices of color j. We consider the perfect 2-colorings of the distance-2 graph of the 24-cube {0, 1}24 with parameters ((20+ c, 256− c)(c, 276− c)) (i.e., with eigenvalue 20). We prove that such colorings exist for all c from 1 t...

We continue the study of the following general problem on vertex colorings of graphs. Suppose that some vertices of a graph G are assigned to some colors. Can this “precoloring” be extended to a proper coloring of G with at most k colors (for some given k)? Here we investigate the complexity status of precoloring extendibility on some classes of perfect graphs, giving good characterizations (ne...