An (r, n)-split coloring of a complete graph is an edge coloring with r colors under which the vertex set is partitionable into r parts so that for each i, part i does not contain K, in color i. This generalizes the notion of split graphs which correspond to (2, 2)-split colorings. The smallest N for which the complete graph Ks has a coloring which is not (r,n)-split is denoted by f,.(n). Balan...

We discuss the following conjecture: If G = (V; E) is a-regular simple graph with an even number of vertices at most 22 then G is edge colorable. In this paper we show that the conjecture is true for large graphs if jV j < (2 ?)). We discuss related results.

In this paper, we propose an algorithm for nding all the edge colorings in bipartite graphs. Our algorithm requires O(T (n;m; ) + Kminfn 2 +m; T (n;m; )g) time and O(m ) space, where n denotes the number of vertices, m denotes the number of edges, denotes the number of maximum degree, T (n;m; ) denotes the time complexity of an edge coloring algorithm, and K denotes the number of edge colorings.

Let Qn be a hypercube of dimension n, that is, a graph whose vertices are binary n-tuples and two vertices are adjacent iff the corresponding n-tuples differ in exactly one position. An edge coloring of a graph H is called rainbow if no two edges of H have the same color. Let f(G, H) be the largest number of colors such that there exists an edge coloring of G with f(G, H) colors such that no su...

An interval coloring of a graph G is a proper coloring of E(G) by positive integers such that the colors on the edges incident to any vertex are consecutive. A (3, 4)-biregular bigraph is a bipartite graph in which each vertex of one part has degree 3 and each vertex of the other has degree 4; it is unknown whether these all have interval colorings. We prove that G has an interval coloring usin...

An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by a(G). From a result of Burnstein it follows that all subcubic graphs are acyclically edge colorable using 5 colors. This result is tight since there are...

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 the Maximal Matching problems, as t...

It is widely accepted that responses from onand o -center cells give rise to edges and are equivalent to edge detectors. In this paper, we point out that onand o -center cell responses provide more information than edges. We show that an edge-based representation makes the ownership of boundaries ambiguous and requires a combinatorial search to model perceptual grouping. By analyzing the di ere...

An edge-coloring of a multigraph G with colors 1, . . . , t is called an interval t-coloring if all colors are used, and the colors of edges incident to any vertex of G are distinct and form an interval of integers. In this note, we show that all Eulerian multigraphs with an odd number of edges have no interval coloring. We also give some methods for constructing of interval non-edge-colorable ...