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 χp(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) ≤ χp(G) ≤ |V (G)| − α(G) + 1, where χ(G) and α(G) are the chromatic(More)
The well-known lower bound on the independence number of a graph due to Caro (New Results on the Independence Number, Technical Report, TelAviv University, 1979) and Wei (A Lower Bound on the Stability Number of a Simple Graph, Technical memorandum, TM 81 11217 9, Bell laboratories, 1981) can be established as a performance guarantee of two natural and(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)
We consider the problem of efficient evacuation using multiple exits. We formulate this problem as a discrete problem on graphs where mobile agents located in distinct nodes of a given graph must quickly reach one of multiple possible exit nodes, while avoiding congestion and bottlenecks. Each node of the graph has the capacity of holding at most one agent(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)
A vertex ranking of a graph G is an assignment of positive integers (colors) to the vertices of G such that each path connecting two vertices of the same color contains a vertex of a higher color. Our main goal is to find a vertex ranking using as few colors as possible. Considering on-line algorithms for vertex ranking of split graphs, we prove that the(More)