We propose a simple algorithm which is based on edge-coloring of system graphs for termination detection of loosely synchronous computations. The proposed algorithm is fully symmetric in that all processors run syntactically identical code and can detect global termination at the same time. Under the 1-port communication model, the algorithm is optimal in terms of termination delay, the diieren...

We investigate the distinguishing index D′(G) of a graph G as the least number d such that G has an edge-colouring with d colours that is only preserved by the trivial automorphism. This is an analog to the notion of the distinguishing number D(G) of a graph G, which is defined for colourings of vertices. We obtain a general upper bound D′(G) ≤ ∆(G) unless G is a small cycle C3, C4 or C5. We al...

A well-known special case of a conjecture attributed to Ryser (actually appeared in the thesis of Henderson [7]) states that k-partite intersecting hypergraphs have transversals of at most k−1 vertices. An equivalent form of the conjecture in terms of coloring of complete graphs is formulated in [1]: if the edges of a complete graph K are colored with k colors then the vertex set of K can be co...

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...