An independent set in an undirected graph G = (V,E) is a set of vertices that induce a subgraph which does not contain any edges. The size of the maximum independent set in G is denoted by α(G). For an integer k, a k-coloring of G is a function σ : V → [1 . . . k] which assigns colors to the vertices of G. A valid k-coloring of G is a coloring in which each color class is an independent set. Th...

A twin edge k-coloring of a graph G is a proper edge k-coloring of G with the elements of Zk so that the induced vertex k-coloring, in which the color of a vertex v in G is the sum in Zk of the colors of the edges incident with v, is a proper vertex k-coloring. The minimum k for which G has a twin edge k-coloring is called the twin chromatic index of G. Twin chromatic index of the square P 2 n ...

We provide a framework for online conflict-free coloring (CF-coloring) of any hypergraph. We use this framework to obtain an efficient randomized online algorithm for CF-coloring any k-degenerate hypergraph. Our algorithm uses O(k log n) colors with high probability and this bound is asymptotically optimal for any constant k. Moreover, our algorithm uses O(k log k log n) random bits with high p...

A distinguishing coloring of a graph G is a coloring of the vertices so that every nontrivial automorphism of G maps some vertex to a vertex with a different color. The distinguishing number of G is the minimum k such that G has a distinguishing coloring where each vertex is assigned a color from {1, . . . , k}. A list assignment to G is an assignment L = {L(v)}v∈V (G) of lists of colors to the...

Consider the following problem: For given graphs G and F1, . . . , Fk, find a coloring of the edges of G with k colors such that G does not contain Fi in color i. For example, if every Fi is the path P3 on 3 vertices, then we are looking for a proper k-edge-coloring of G, i.e., a coloring of the edges of G with no pair of edges of the same color incident to the same vertex. Rödl and Ruciński st...

A vertex-distinguishing coloring of a graph G consists in an edge or a vertex coloring (not necessarily proper) of G leading to a labeling of the vertices of G, where all the vertices are distinguished by their labels. There are several possible rules for both the coloring and the labeling. For instance, in a set irregular edge coloring [5], the label of a vertex is the union of the colors of i...

A total coloring of a graph G is a coloring of all elements of G, i.e. vertices and edges, in such a way that no two adjacent or incident elements receive the same color. The total coloring problem is to find a total coloring of a given graph with the minimum number of colors. Many combinatorial problems can be efficiently solved for partial k-trees, i.e., graphs with bounded tree-width. Howeve...

Let c be a vertex k -coloring on a connected graph G(V,E) . Let Π = {C1, C2, ..., Ck} be the partition of V (G) induced by the coloring c . The color code cΠ(v) of a vertex v in G is (d(v, C1), d(v, C2), ..., d(v, Ck)), where d(v, Ci) = min{d(v, x)|x ∈ Ci} for 1 ≤ i ≤ k. If any two distinct vertices u, v in G satisfy that cΠ(u) 6= cΠ(v), then c is called a locating k-coloring of G . The locatin...