Piotr Borowiecki

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A parity path in a vertex colouring of a graph is a path along which each colour is used an even number of times. Let χ p (G) be the least number of colours in a proper vertex colouring of G having no parity path. It is proved that for any graph G we have the following tight bounds χ(G) ≤ χ p (G) ≤ |V (G)| − α(G) + 1, where χ(G) and α(G) are the chromatic(More)
Dynamics is an inherent feature of many real life systems so it's natural to define and investigate the properties of models that reflect dynamic aspects of systems. In this talk we investigate the dynamic approach to the problem of graph coloring, which is well known and widely used in system modeling. In the dynamic setting of the problem, the graph we(More)
The well-known lower bound on the independence number of a graph due 1981) can be established as a performance guarantee of two natural and simple greedy algorithms or of a simple randomized algorithm. We study possible generalizations and improvements of these approaches using vertex weights and discuss conditions on so-called potential functions p G : V(More)
In this work we consider the edge searching problem for vertex-weighted graphs with arbitrarily fast and invisible fugitive. The weight function $${\omega }$$ ω provides for each vertex $$v$$ v the minimum number of searchers required to guard $$v$$ v , i.e., the fugitive may not pass through $$v$$ v without being detected only if at least $${\omega }(v)$$(More)
In graph cleaning problems, brushes clean a graph by traversing it subject to certain rules. We consider the process where at each time step, a vertex that has at least as many brushes as incident, contaminated edges, sends brushes down these edges to clean them. Various problems arise, such as determining the minimum number of brushes (called the brush(More)