Marek Szykuła

Rainbow Induced Subgraphs in Proper Vertex Colorings

Given a graph H we define ρ(H) to be the minimum order of a graph G such that every proper vertex coloring of G contains a rainbow induced subgraph isomorphic to H. We give upper and lower bounds for ρ(H), compute the exact value for some classes of graphs, and consider an interesting combinatorial problem connected with computation of ρ(H) for paths. A part of this research has been guided by a computer search and, accordingly, some computational results are presented. A special motivation comes from research in on-line coloring.

Computing the shortest reset words of synchronizing automata

In this paper we give the details of our new algorithm for finding minimal reset words of finite synchronizing automata. The problem is known to be computationally hard, so our algorithm is exponential in the worst case, but it is faster than the algorithms used so far and it performs well on average. The main idea is to use a bidirectional breadth-first-search and radix (Patricia) tries to store and compare subsets. A good performance is due to a number of heuristics we apply and describe here in a suitable detail. We give both theoretical and practical arguments showing that the effective branching factor is considerably reduced. As a practical test we perform an experimental study of the length of the shortest reset word for random automata with up to

Generating small automata and the Černý conjecture

n=350

A Fast Algorithm Finding the Shortest Reset Words

n=350 states and up to

On the State Complexity of the Shuffle of Regular Languages

k=10

Algebraic Synchronization Criterion and Computing Reset Words

k=10 input letters. In particular, we obtain a new estimation of the expected length of the shortest reset word