Andrzej Grzesik

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An edge-coloring of a graph G with colors 1,. .. , t is an interval t-coloring if all colors are used, and the colors of edges incident to each vertex of G are distinct and form an interval of integers. A graph G is interval colorable if it has an interval t-coloring for some positive integer t. In this note we prove that K 1,m,n is interval colorable if(More)
If a graph G has n ≥ 4k vertices and more than n 2 /4 edges, then it contains a copy of C 2k+1. But can we guarantee something stronger than just a single copy? In 1992, Erd˝ os, Faudree and Rousseau showed that the number of edges that occur in a triangle of such a G is at least 2 n/2 + 1, and this bound is tight. They also showed that the minimum number(More)
In this paper, we study (1 : b) Avoider-Enforcer games played on the edge set of the complete graph on n vertices. For every constant k ≥ 3 we analyse the k-star game, where Avoider tries to avoid claiming k edges incident to the same vertex. We analyse both versions of Avoider-Enforcer games – the strict and the monotone – and for each provide explicit(More)
We examine the evolution of the best choice algorithm and the probability of its success from a directed path to the linear order of the same cardinality through kth powers of a directed path, 1 ≤ k < n. The vertices of a kth power of a directed path of a known length n are exposed one by one to a selector in some random order. At any time the selector can(More)
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