An acyclic edge-coloring of a graph is a proper edge-coloring without bichromatic (2colored) cycles. The acyclic chromatic index of a graph G, denoted by a(G), is the least integer k such that G admits an acyclic edge-coloring using k colors. Let ∆ = ∆(G) denote the maximum degree of a vertex in a graph G. A complete bipartite graph with n vertices on each side is denoted by Kn,n. Basavaraju, C...

Given a graph G = (V,E) and positive integral vertex weights w : V → N, the max-coloring problem seeks to find a proper vertex coloring of G whose color classes C1, C2, . . . , Ck, minimize ∑k i=1 maxv∈Ciw(v). The problem arises in scheduling conflicting jobs in batches and in minimizing buffer size in dedicated memory managers. In this paper we present three approximation algorithms and one in...

A strong edge-coloring of a graph is a function that assigns to each edge a color such that every two distinct edges that are adjacent or adjacent to a same edge receive different colors. The strong chromatic index χs(G) of a graph G is the minimum number of colors used in a strong edge-coloring of G. From a primal-dual point of view, there are three natural lower bounds of χs(G), that is σ(G) ...

An edge coloring of a graph G is a process of assigning colors to the adjacent edges so that the adjacent edges represents the different colors. In this paper, an algorithm is proposed to find the perfect color matching of the regular bipartite multigraph with low time complexity. For that, the proposed algorithm is divided into two procedures. In the first procedure, the possible circuits and ...

Given a graph G = (V, E) and a spanning subgraph H of G (the backbone of G), a backbone coloring for G and H is a proper vertex coloring V → {1, 2, . . .} of G in which the colors assigned to adjacent vertices in H differ by at least two. In a recent paper, backbone colorings were introduced and studied in cases were the backbone is either a spanning tree or a spanning path. Here we study the c...

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...

A coloring of the q-ary n-dimensional cube (hypercube) is called perfect if, for every n-tuple x, the collection of the colors of the neighbors of x depends only on the color of x. A Boolean-valued function is called correlation-immune of degree n − m if it takes the value 1 the same number of times for each m-dimensional face of the hypercube. Let f = χ be a characteristic function of some sub...

In this paper, we consider the problem of finding an edge coloring of a d-regular bipartite multigraph with 2n vertices and m = nd edges. The best known deterministic algorithm (by Cole, Ost, and Schirra) takes O(m log d) time to find an edge coloring of such a graph. This bound is achieved by combining an O(m)-time perfect-matching algorithm with the Euler partition algorithm. The O(m) time bo...