Galvin ([7]) proved that every k-edge-colorable bipartite multigraph is kedge-choosable. Slivnik ([11]) gave a streamlined proof of Galvin's result. A multigraph G is said to be nearly bipartite if it contains a special vertex Vs such that G Vs is a bipartite multigraph. We use the technique in Slivnik's proof to obtain a list coloring analog of Vizing's theorem ([12]) for nearly bipartite mult...

Graph coloring is a central problem in distributed computing. Both vertexand edge-coloring problems have been extensively studied in this context. In this paper we show that a (2∆ − 1)-edge-coloring can be computed in time smaller than log n for any > 0, specifically, in e √ log logn) rounds. This establishes a separation between the (2∆ − 1)-edge-coloring and Maximal Matching problems, as the ...

An additive hereditary property of graphs is a class of simple graphs which is closed under unions, subgraphs and isomorphism. Let Q be an additive hereditary property of graphs. A Q-edge-coloring of a simple graph is an edge coloring in which the edges colored with the same color induce a subgraph of property Q. In this paper we present some results on fractional Q-edge-colorings. We determine...

A twin edge k-coloring of a graph G is a proper edge k-coloring of G with the elements of Zk so that the induced vertex k-coloring, in which the color of a vertex v in G is the sum in Zk of the colors of the edges incident with v, is a proper vertex k-coloring. The minimum k for which G has a twin edge k-coloring is called the twin chromatic index of G. Twin chromatic index of the square P 2 n ...

Abstract. In a multi-channel Wireless Mesh Networks (WMN), each node is able to use multiple non-overlapping frequency channels. Raniwala et al. (Mobile Computing and Communications Review 2004, INFOCOM 2005) propose and study several such architectures in which a computer can have multiple network interface cards. These architectures are modeled as a graph problem named maximum edge q-coloring...

An interval edge t − coloring of a graph G is a proper edge coloring of G with colors 1, 2, , t ... such that at least one edge of G is colored by color , 1, 2, , i i t = ... , and the edges incident with each vertex ( ) v V G ∈ are colored by ( ) G d v consecutive colors, where ( ) G d v is the degree of the vertex v in G . In this paper interval edge colorings of bipartite cylinders and bipar...

The problem of edge-coloring a bipartite graph is to color the edges so that adjacent edges receive di erent colors. An optimal algorithm uses the minimum number of colors to color the edges. We consider several approximation algorithms for edge-coloring bipartite graphs and show tight bounds on the number of colors they use in the worst case. We also brie y consider the constrained edge-colori...

A 1-plane graph is a graph embedded in the plane such that each edge is crossed at most once. A 1-plane graph is optimal if it has maximum edge density. A red-blue edge coloring of an optimal 1-plane graph G partitions the edge set of G into blue edges and red edges such that no two blue edges cross each other and no two red edges cross each other. We prove the following: (i) Every optimal 1-pl...

Brualdi and Hollingsworth conjectured that, for even n, in a proper edge coloring of Kn using precisely n − 1 colors, the edge set can be partitioned into n2 spanning trees which are rainbow (and hence, precisely one edge from each color class is in each spanning tree). They proved that there always are two edge disjoint rainbow spanning trees. Kaneko, Kano and Suzuki improved this to three edg...

Given an n-vertex graph G, an edge-coloring of G with natural numbers is a consecutive (or interval) coloring if the colors of edges incident with each vertex are distinct and form an interval of integers. In this paper we prove that if G has a consecutive coloring and n¿3 then S(G)62n − 4, where S(G) is the maximum number of colors allowing a consecutive coloring. Next, we investigate the so-c...