Konrad Piwakowski

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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)
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)
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)
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)