For a nontrivial connected graph G, let c : V (G) → N be a vertex coloring of G where adjacent vertices may be colored the same. For a vertex v of G, the neighborhood color set NC(v) is the set of colors of the neighbors of v. The coloring c is called a set coloring if NC(u) 6= NC(v) for every pair u, v of adjacent vertices of G. The minimum number of colors required of such a coloring is calle...

We are given a edge-weighted undirected graph G = (V,E) and a set of labels/colors C = {1, 2, . . . p}. A nonempty subset Cv ⊆ C is associated with each vertex v ∈ V . A coloring of the vertices is feasible if each vertex v is colored with a color of Cv. A coloring uniquely defines a subset E ′ ⊆ E of edges having different colored endpoints. The problem of finding a feasible coloring which def...

In 1998, Karpovsky, Chakrabarty and Levitin introduced identifying codes to model fault diagnosis in multiprocessor systems [1]. In these codes, each vertex is identified by the vertices belonging to the code in its neighborhood. There exists a coloring variant as follows: a globally identifying coloring of a graph is a coloring such that each vertex is identified by the colors in its neighborh...

A proper coloring of a graph is odd if every non-isolated vertex has some color that appears an number times on its neighborhood. This notion was recently introduced by Petruševski and Škrekovski, who proved planar admits 9-coloring; they also conjectured 5-coloring. Shortly after, this conjecture confirmed for graphs girth at least seven Cranston; outerplanar Caro, Petruševski, Škrekovski. Bui...

Given a graph G = (V,E) with n vertices, m edges and maximum vertex degree ∆, the load distribution of a coloring φ : V → {red, blue} is a pair dφ = (rφ, bφ), where rφ is the number of edges with at least one end-vertex colored red and bφ is the number of edges with at least one end-vertex colored blue. Our aim is to find a coloring φ such that the (maximum) load, lφ := 1 m · max{rφ, bφ}, is mi...

A k-list assignment L of a graph G is a mapping which assigns to each vertex v of G a set L(v) of size k. A (k,t)-list assignment of G is a k-list assignment with | ⋃ v∈V (G) L(v)| = t. An L-coloring φ of G is a proper coloring of G such that φ(v) is chosen from L(v) for every vertex v. A graph G is Lcolorable if G has an L-coloring. When the parameter t is not of special interest, we simply sa...

We study weighted bipartite edge coloring problem, which is a generalization of two classical problems: bin packing and edge coloring. This problem has been inspired from the study of Clos networks in multirate switching environment in communication networks. In weighted bipartite edge coloring problem, we are given an edge-weighted bipartite multigraph G = (V,E) with weights w : E → [0, 1]. Th...

This lecture deals with the problem of proper vertex colorings of graphs. More speciically, we are interested in coloring a given 3-colorable graph with as few colors as we can. It is easy to see that it is NP-Hard to nd a 3 coloring for any given 3-colorable graph-this would enable deciding whether a general graph is 3-colorable. If a polynomial-time algorithm existed for 3-coloring a 3-colora...

Let r,k?1 be two integers. An r-hued k-coloring of the vertices a graph G=(V,E) is proper vertices, such that, for every vertex v?V, number colors in its neighborhood at least min{dG(v),r}, where dG(v) degree v. We prove existence an (r+1)-coloring planar graphs with girth 8 r?9. As corollary, maximum ??9 and admits 2-distance (?+1)-coloring.