Introduction to finite fields and their applications: Preface

We continue the study of the following general problem on the vertex colourings of graphs. Suppose that some vertices of a graph G are assigned to some colours. Can this ‘precolouring’ be extended to a proper colouring of G with at most k colours (for some given k)? Here we investigate the complexity status of precolouring extendibility on some classes of perfect graphs, giving good characterizations (necessary and sufficient conditions) that lead to algorithms with linear or polynomial running time. It is also shown how a larger subclass of perfect graphs can be derived from graphs containing no induced path on four vertices.

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Precoloring Extension III: Classes of Perfect Graphs

On zero-divisor graphs of finite rings

Monotone Boolean functions are an important object in discrete mathematics and mathematical cybernetics. Topics related to these functions have been actively studied for several decades. Many results have been obtained, and many papers published. However, until now there has been no sufficiently complete monograph or survey of results of investigations concerning monotone Boolean functions. The object of this survey is to present the main results on monotone Boolean functions obtained during the last 50 years.

Monotone Boolean functions

A refinement of an algorithm developed by Culberson and Rawlins yields the numbers of all partially ordered sets (posets) with n points and k antichains for n≤11 and all relevant integers k. Using these numbers in connection with certain formulae derived earlier by the first author, one can now compute the numbers of all quasiordered sets, posets, connected posets etc. with n points for n≤14. Using the well-known one-to-one correspondence between finite quasiordered sets and finite topological spaces, one obtains the numbers of finite topological spaces with n points and k open sets for n≤11 and all k, and then the numbers of all topologies on n≤14 points satisfying various degrees of separation and connectedness properties, respectively. The number of (connected) topologies on 14 points exceeds 1023.

Counting finite posets and topologies

Symmetry Groups of Boolean Functions and Constructions of Permutation Groups

A clone C on a set A is a set of operations on A containing the projection operations and closed under composition. A combinatorial invariant of a clone is its pn-sequence (po0(C), PI (C), . . . ) , where p,(C) is the number of essentially n-ary operations in C. We investigate the links between this invariant and structural properties of clones. It has been conjectured that the pn-sequence of a clone on a finite set is either eventually strictly increasing or is bounded above by a finite constant. We verify this conjecture for.a large family of clones. A special role in our work is played by totally symmetric operations and totally symmetric clones. We show that every totally symmetric clone on a finite set has a bounded pn-sequence and that it is decidable if a clone is totally symmetric.

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On the number of operations in a clone

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 $$n=350$$ n = 350 states and up to $$k=10$$ k = 10 input letters. In particular, we obtain a new estimation of the expected length of the shortest reset word $${\approx \,}2.5\sqrt{n-5}$$ � 2.5 n - 5 for binary automata and show that the error of this estimate is sufficiently small. Experiments for automata with more than two input letters show certain trends with the same general pattern.

https://link.springer.com/content/pdf/10.1007%2Fs10878-013-9682-0.pdf

Computing the shortest reset words of synchronizing automata

The Cerný's conjecture states that for every synchronizing automaton with n states there exists a reset word of length not exceeding (n - 1)2. We prove this conjecture for a class of automata preserving certain properties of intervals of a directed graph. Our result unifies and generalizes some earlier results obtained by other authors.

https://hal.inria.fr/hal-00966381/document

Andrzej Kisielewicz

Papers

Symmetry Groups of Boolean Functions and Constructions of Permutation Groups

On the number of operations in a clone

Computing the shortest reset words of synchronizing automata

The Cerný conjecture for automata respecting intervals of a directed graph

A solution of Dedekind's problem on the number of isotone Boolean functions.