A k-edge-weighting w of a graph G is an assignment of an integer weight, w(e) ∈ {1, . . . , k}, to each edge e. An edge weighting naturally induces a vertex coloring c by defining c(u) = ∑ u∼e w(e) for every u ∈ V (G). A k-edge-weighting of a graph G is vertexcoloring if the induced coloring c is proper, i.e., c(u) ≠ c(v) for any edge uv ∈ E(G). Given a graph G and a vertex coloring c0, does th...

We prove that proper coloring distinguishes between block-factors and finitely dependent stationary processes. A stochastic process is finitely dependent if variables at sufficiently wellseparated locations are independent; it is a block-factor if it can be expressed as a finite-range function of independent variables. The problem of finding non-block-factor finitely dependent processes dates b...

Let G be a graph with no isolated vertices. A k-coupon coloring of G is an assignment of colors from [k] := {1, 2, . . . , k} to the vertices of G such that the neighborhood of every vertex of G contains vertices of all colors from [k]. The maximum k for which a k-coupon coloring exists is called the coupon coloring number of G, and is denoted χc(G). In this paper, we prove that every d-regular...

For integers k > 0 and r > 0, a conditional (k, r)-coloring of a graph G is a proper k-coloring of the vertices of G such that every vertex v of degree d(v) in G is adjacent to vertices with at least min{r, d(v)} different colors. The smallest integer k for which a graph G has a conditional (k, r)-coloring is called the rth order conditional chromatic number, denoted by χr(G). For different val...

The Coloring problem is to test whether a given graph can be colored with at most k colors for some given k, such that no two adjacent vertices receive the same color. The complexity of this problem on graphs that do not contain some graph H as an induced subgraph is known for each fixed graph H. A natural variant is to forbid a graph H only as a subgraph. We call such graphs strongly H-free an...

A graph G is equitably k-colorable if its vertices can be partitioned into k independent sets in such a way that the number of vertices in any two sets differ by at most one. The smallest integer k for which such a coloring exists is known as the equitable chromatic number of G and it is denoted by χ=(G). In this paper the problem of determinig the value of equitable chromatic number for multic...

Given a graph G= (V,E), a greedy coloring of G is a proper coloring such that, for each two colors i< j, every vertex of V (G) colored j has a neighbor with color i. The greatest k such that G has a greedy coloring with k colors is the Grundy number of G. A b-coloring of G is a proper coloring such that every color class contains a vertex which is adjacent to at least one vertex in every other ...

One way to state the Load Coloring Problem (LCP) is as follows. Let G = (V,E) be graph and let f : V → {red,blue} be a 2-coloring. An edge e ∈ E is called red (blue) if both end-vertices of e are red (blue). For a 2-coloring f , let r′ f and b′f be the number of red and blue edges and let μf (G) = min{r ′ f , b ′ f}. Let μ(G) be the maximum of μf (G) over all 2-colorings. We introduce the param...

Let G = G(n) be a graph on n vertices with maximum degree ∆ = ∆(n). Assign to each vertex v of G a list L(v) of colors by choosing each list independently and uniformly at random from all k-subsets of a color set C of size σ = σ(n). Such a list assignment is called a random (k, C)-list assignment. In this paper, we are interested in determining the asymptotic probability (as n → ∞) of the exist...

Let G = (V,E) be a nontrivial, finite, and connected graph. A function c from E to {1,2,...,k},k ∈ N, can considered as rainbow k-coloring if every two vertices x y in has an x- path. Therefore, no path's edges receive the same color; this condition is called “rainbow path”. The smallest positive integer k, designated by rc(G), connection number. Thus, k-coloring. Meanwhile, strong within for h...