Let G be a graph whose each component has order at least 3. Let s : E(G) → Zk for some integer k ≥ 2 be an improper edge coloring of G (where adjacent edges may be assigned the same color). If the induced vertex coloring c : V (G) → Zk defined by c(v) = ∑ e∈Ev s(e) in Zk, (where the indicated sum is computed in Zk and Ev denotes the set of all edges incident to v) results in a proper vertex col...

A linear equation L is called k-regular if every k-coloring of the positive integers contains a monochromatic solution to L. Richard Rado conjectured that for every positive integer k, there exists a linear equation that is (k − 1)-regular but not k-regular. We prove this conjecture by showing that the equation Pk−1 i=1 2 2i−1xi = “ −1 + Pk−1 i=1 2 2i−1 ” x0 has this property. This conjecture i...

Ž Many combinatorial problems can be efficiently solved for partial k-trees graphs . of treewidth bounded by k . The edge-coloring problem is one of the well-known combinatorial problems for which no efficient algorithms were previously known, except a polynomial-time algorithm of very high complexity. This paper gives a linear-time sequential algorithm and an optimal parallel algorithm which f...

Given a graph G, by a Grundy k-coloring of G we mean any proper k-vertex coloring of G such that for each two colors i and j, i < j , every vertex ofG colored by j has a neighbor with color i. The maximum k for which there exists a Grundy k-coloring is denoted by (G) and called Grundy (chromatic) number of G. We first discuss the fixed-parameter complexity of determining (G) k, for any fixed in...

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 Δ≥9 and admits 2-distance (Δ+1)-coloring.

Given a graph G and list assignment L(v) for each vertex of v G, proper L-list-coloring is function that maps every to color in such no pair adjacent vertices have the same color. We say k-list-colorable when has colors size at least k. A 2-distance coloring where distance most 2 cannot share prove existence ( $$\Delta +2$$ )-coloring planar graphs with girth 10 maximum degree \ge 4$$ .

We prove that for every fixed k and ` ≥ 5 and for sufficiently large n, every edge coloring of the hypercube Qn with k colors contains a monochromatic cycle of length 2`. This answers an open question of Chung. Our techniques provide also a characterization of all subgraphs H of the hypercube which are Ramsey, i.e., have the property that for every k, any k-edge coloring of a sufficiently large...

We consider a coloring problem on dynamic, one-dimensional point sets: points appearing and disappearing on a line at given times. We wish to color them with k colors so that at any time, any sequence of p(k) consecutive points, for some function p, contains at least one point of each color. We prove that no such function p(k) exists in general. However, in the restricted case in which points a...

Some situations concerning cost allocation are formulated as combinatorial optimization games. We consider a minimum coloring game and a minimum vertex cover game. For a minimum coloring game, Deng{Ibaraki{Nagamochi 1] showed that deciding the core nonemptiness of a given minimum coloring game is NP-complete, which implies that a good characterization of balanced minimum coloring games is unlik...