In the (k, i)-coloring problem, we aim to assign sets of colors of size k to the vertices of a graph G, so that the sets which belong to adjacent vertices of G intersect in no more than i elements and the total number of colors used is minimum. This minimum number of colors is called the (k, i)-chromatic number. We present in this work a very simple linear time algorithm to compute an optimum (...

Let S be a point set in general position on the plane such that its elements are colored red or blue. We study the following problem: Remove as few points as possible from S such that the remaining points can be enclosed by two isothetic rectangles, one containing all the red points, the other all the blue points, and such that each rectangle contains only points of one color. We prove that thi...

An acyclic coloring of a graph G is a coloring of its vertices such that: (i) no two neighbors in G are assigned the same color and (ii) no bicolored cycle can exist in G. The acyclic chromatic number of G is the least number of colors necessary to acyclically color G. In this paper, we show that any graph of maximum degree 5 has acyclic chromatic number at most 9, and we give a linear time alg...

A coloring of a graph G = (V ,E) is a partition {V1, V2, . . . , Vk} of V into independent sets or color classes. A vertex v ∈ Vi is a Grundy vertex if it is adjacent to at least one vertex in each color class Vj for every j <i. A coloring is a Grundy coloring if every color class contains at least one Grundy vertex, and the Grundy number of a graph is the maximum number of colors in a Grundy c...

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...

A proper vertex coloring of a simple graph is $k$-forested if the graph induced by the vertices of any two color classes is a forest with maximum degree less than $k$. A graph is $k$-forested $q$-choosable if for a given list of $q$ colors associated with each vertex $v$, there exists a $k$-forested coloring of $G$ such that each vertex receives a color from its own list. In this paper, we prov...

Many combinatorial problems can be efficiently solved for series]parallel multigraphs. However, the edge-coloring problem of finding the minimum number of colors required for edge-coloring given graphs is one of a few well-known combinatorial problems for which no efficient algorithms have been obtained for series]parallel multigraphs. This paper gives a linear algorithm for the problem on seri...

3 (m,p,c)-sets 59 4 Canonical Results 63 5 Coloring Objects of Higher Rank 65 1 Introduction In 1930 Ramsey published his paper On a problem in formal logic 12]. He established a result, nowadays known as Ramsey's Theorem: Let k and r be positive integers. Then for every r-coloring of the k-element subsets of ! there exists an innnite subset S ! such that all k-element subsets of S are colored ...

For integers k and l, each greater than 1, suppose that p is a prime with p ≡ 1 (mod k) and that the kth-power classes mod p induce a coloring of the integer segment [0, p− 1] that admits no monochromatic occurrence of l consecutive members of an arithmetic progression. Such a coloring can lead to a coloring of [0, (l − 1)p] that is similarly free of monochromatic l-progressions, and, hence, ca...

A problem of a visual image of a directed finite graph has appeared in the study of the road coloring conjecture. Given a finite directed graph, a coloring of its edges turns the graph into finite-state automaton. The visual perception of the structure properties of automata is an important goal. A synchronizing word of a deterministic automaton is a word in the alphabet of colors of its edges ...