Konrad Piwakowski

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For given graphs G1, G2, ..., Gk, where k ≥ 2, the multicolor Ramsey number R(G1, G2, ..., Gk) 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 Gi colored with i, for some 1 ≤ i ≤ k. Let Pk (resp. Ck) be the path (resp. cycle) on k vertices. In(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)
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-tocolor (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)
This paper investigates the complexity of scheduling biprocessor 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)