In this note an adaptation of heuristic tabu search algorithm for nding Ramsey graphs is presented. As a result, seven new lower bounds for classical Ramsey numbers are established:
We show that, in any coloring of the edges of K 38 with two colors, there exists a triangle in the first color or a monochromatic K 10 −e (K 10 with one edge removed) in the second color, and hence we obtain a bound on the corresponding Ramsey number, R(K 3 , K 10 −e) ≤ 38. The new lower bound of 37 for this number is established by a coloring of K 36… (More)
For a given approximate vertex coloring algorithm a graph is said to be slightly hard-to-color (SHC) if some implementation of the algorithm uses more colors than the minimum needed. Similarly, a graph is said to be hard-to-color (HC) if every implementation of the algorithm results in a nonoptimal coloring. We study smallest such graphs for the… (More)
We consider the problem of efficient coloring of the edges of a so-called binomial tree T, i.e. acyclic graph containing two kinds of edges: those which must have a single color and those which are to be colored with L consecutive colors, where L is an arbitrary integer greater than 1. We give an O(n) time algorithm for optimal coloring of such a tree,… (More)
is the smallest integer n such that if we arbitrarily color the edges of the complete graph on n vertices with k colors, there is always a monochromatic copy of G i colored with i, for some 1 ≤ i ≤ k. Let P k (resp. C k) be the path (resp. cycle) on k vertices. In the paper we show that R(P 3 , C k , C k) = R(C k , C k) = 2k − 1 for odd k. In addition, we… (More)
This paper investigates the complexity of scheduling bipro-cessor tasks on dedicated processors to minimize mean flow time. Since the general problem is strongly NP-hard, we assume some restrictions on task lengths and the structure of associated scheduling graphs. Of particular interest are acyclic graphs. In this way we identify a borderline between… (More)
For a given approximate coloring algorithm a graph is said to be slightly hard-to-color (SHC) if some implementation of the algorithm uses more colors than the chromatic number. Similarly, a graph is said to be hard-to-color (HC) if every implementation of the algorithm results in a non-optimal coloring. In the paper, we study the smallest of such graphs… (More)
With the help of computer algorithms, we improve the lower bound on the Ramsey multiplicity of K4, and thus show that the exact value of it is equal to 9.