We further develop the 0/1 ILP formulation of Lee for edge coloring where colors are encoded in binary. With respect to that formulation, our main contributions are: (i) an efficient separation algorithm for general block inequalities, (ii) an efficient LP-based separation algorithm for stars (i.e., the all-different polytope), (iii) introduction of matching inequalities, and (iv) introduction ...

A coloring of the edges of Kn is constructed such that every copy of K4 has at least three colors on its edges. As n → ∞, the number of colors used is eO( √ log n ). This improves upon the previous probabilistic bound of O( √ n ) due to Erdős and Gyárfás.

Let G and H be two graphs. A proper vertex coloring of G is called a dynamic coloring, if for every vertex v with degree at least 2, the neighbors of v receive at least two different colors. The smallest integer k such that G has a dynamic coloring with k colors denoted by χ2(G). We denote the cartesian product of G and H by G¤H. In this paper, we prove that if G and H are two graphs and δ(G) ≥...

A graph G with maximum degree and edge chromatic number ′(G)> is edge-critical if ′(G− e)= for every edge e of G. New lower bounds are given for the average degree of an edge-critical graph, which improve on the best bounds previously known for most values of . Examples of edge-critical graphs are also given. In almost all cases, there remains a large gap between the best lower bound known and ...

We investigate structural properties of planar graphs without triangles or without 4-cycles, and show that every triangle-free planar graph G is edge-( (G) + 1)-choosable and that every planar graph with (G) = 5 and without 4-cycles is also edge-( (G) + 1)-choosable. c © 2003 Elsevier B.V. All rights reserved.

We shall use the distance chromatic index defined by the present author in early nineties, cf. [5] or [4] of 1993. The edge distance of two edges in a multigraph M is defined to be their distance in the line graph L(M) of M . Given a positive integer d, define the d+-chromatic index of the multigraph M , denoted by q(d)(M), to be equal to the chromatic number χ of the dth power of the line grap...

A nearly equitable edge-coloring of a multigraph is a coloring such that edges incident to each vertex are colored equitably in number. This problem was solved in O(kn2) time, where n and k are the numbers of the edges and the colors, respectively. The running time was improved to be O(n2/k + n|V |) later. We present a more efficient algorithm for this problem that runs in O(n2/k) time. key wor...

We consider the problem of coloring edges of a graph subject to the following constraints: for every vertex v, all the edges incident with v have to be colored with at most q colors. The goal is to find a coloring satisfying the above constraints and using the maximum number of colors. Notice that the notion of coloring is different than in the classical edge coloring problem, as neighboring ed...

For two graphs, G and H , an edge coloring of a complete graph is (G,H)-good if there is no monochromatic subgraph isomorphic to G and no rainbow subgraph isomorphic to H in this coloring. The set of numbers of colors used by (G,H)-good colorings of Kn is called a mixed Ramsey spectrum. This note addresses a fundamental question of whether the spectrum is an interval. It is shown that the answe...

We produce an edge-coloring of the complete 3-uniform hypergraph on n vertices with e √ log logn) colors such that the edges spanned by every set of five vertices receive at least three distinct colors. This answers the first open case of a question of Conlon-Fox-Lee-Sudakov [1] who asked whether such a coloring exists with (log n) colors.