We investigate the computational complexity of the following problem. We are given a graph in which each vertex has an initial and a target color. Each pair of adjacent vertices can swap their current colors. Our goal is to perform the minimum number of swaps so that the current and target colors agree at each vertex. When the colors are chosen from {1, 2, . . . , c}, we call this problem c-Col...

Chung and Liu defined the d-chromatic Ramsey numbers as a generalization of Ramsey numbers by replacing a weaker condition. Let 1 < d < c and let t = (c d ) . Assume A1, A2, . . . , At are all d-subsets of a set containing c distinct colors. Let G1, G2, . . . , Gt be graphs. The d-chromatic Ramsey number denoted by rc d(G1, G2, . . . , Gt) is defined as the least number p such that, if the edge...

Let C be a set of colors, and let ω(c) be an integer cost assigned to a color c in C. An edge-coloring of a graph G is to color all the edges of G so that any two adjacent edges are colored with different colors in C. The cost ω( f ) of an edge-coloring f of G is the sum of costs ω( f (e)) of colors f (e) assigned to all edges e in G. An edge-coloring f of G is optimal if ω( f ) is minimum amon...

Given a c-edge-colored graph G on n vertices, we define the degree matrix M as the c× n matrix whose entry dij is the degree of color i at vertex vj . We show that the obvious necessary conditions for a c× n matrix to be the degree matrix of a c-edge-colored forest on n vertices are also sufficient. It is well known that non-negative integers d1, d2, . . . , dn form a degree list of a forest on...

A k-edge-coloring of a graph G = (V, E) is a function c that assigns an integer c(e) (called color) in {0, 1, · · · , k−1} to every edge e ∈ E so that adjacent edges get different colors. A k-edge-coloring is linear compact if the colors incident to every vertex are consecutive. The problem k − LCCP is to determine whether a given graph admits a linear compact k-edge coloring. A k-edge-coloring...

Let C be a set of colors, and let ω be a cost function which assigns a real number ω(c) to each color c in C. An edge-coloring of a graph G is to color all the edges of G so that any two adjacent edges are colored with different colors. In this paper we give an efficient algorithm to find an optimal edge-coloring of a given tree T , that is, an edgecoloring f of T such that the sum of costs ω(f...

In this article we introduce (combinatorial) multi{color discrepancy and generalize some classical results from 2{color discrepancy theory to c colors. We give a recursive method that constructs c{colorings from approximations to the 2{color discrepancy. This method works for a large class of theorems like the six{standard{deviation theorem of Spencer, the Beck{Fiala theorem and the results of ...