We prove that for every fixed k and ` ≥ 5 and for sufficiently large n, every edge coloring of the hypercube Qn with k colors contains a monochromatic cycle of length 2`. This answers an open question of Chung. Our techniques provide also a characterization of all subgraphs H of the hypercube which are Ramsey, i.e., have the property that for every k, any k-edge coloring of a sufficiently large...

Consider a graph G with chromatic number k and a collection of complete bipartite graphs, or bicliques, that cover the edges of G. We prove the following two results: • If the bipartite graphs form a partition of the edges of G, then their number is at least 2 √ log2 . This is the first improvement of the easy lower bound of log2 k, while the Alon-Saks-Seymour conjecture states that this can be...

In this paper we introduce a new algorithm for secondary school timetabling, inspired by the classical bipartite graph edge colouring algorithm for basic class-teacher timetabling. We give practical methods for generating large sets of meetings that can be timetabled to run simultaneously, and for building actual timetables based on these sets. We report promising empirical results for one real...

A strong edge-coloring of a graph is a function that assigns to each edge a color such that two edges within distance two apart receive different colors. The strong chromatic index of a graph is the minimum number of colors used in a strong edge-coloring. This paper determines strong chromatic indices of cacti, which are graphs whose blocks are cycles or complete graphs of two vertices. The pro...

Let G be an edge-colored graph. A heterochromatic path of G is such a path in which no two edges have the same color. dc(v) denotes the color degree of a vertex v of G. In a previous paper, we showed that if dc(v) ≥ k for every vertex v of G, then G has a heterochromatic path of length at least dk+1 2 e. It is easy to see that if k = 1, 2, G has a heterochromatic path of length at least k. Sait...

In an ordinary edge-coloring of a graph each color appears at each vertex at most once. An f -coloring is a generalized edge-coloring in which each color appears at each vertex v at most f(v) times where f(v) is a positive integer assigned to v. This paper gives efficient sequential and parallel algorithms to find ordinary edge-colorings and f -colorings for various classes of graphs such as bi...

An adjacent vertex distinguishing edge-coloring of a simple graph G is a proper edge-coloring of G such that no pair of adjacent vertices meets the same set of colors. The minimum number of colors χa(G) required to give G an adjacent vertex distinguishing coloring is studied for graphs with no isolated edge. We prove χa(G) ≤ 5 for such graphs with maximum degree Δ(G) = 3 and prove χa(G) ≤ Δ(G) ...

There are five possible structures for a set of three lines of a Steiner triple system. Each of these three-line "configurations" gives rise to a colouring problem in which a partition of all the lines of an STS( v) is sought, the components of the partition each having the property of not containing any copy of the configuration in question. For a three-line configuration B, and STS ( v) S, th...

Let G be a connected graph of size at least 2 and c :E(G)→{0, 1, . . . , k− 1} an edge coloring (or labeling) of G using k labels, where adjacent edges may be assigned the same label. For each vertex v of G, the color code of v with respect to c is the k-vector code(v) = (a0, a1, . . . , ak−1), where ai is the number of edges incident with v that are labeled i for 0 ≤ i ≤ k − 1. The labeling c ...