Related papers: The \v{C}erny conjecture
The results of several papers concerning the \v{C}ern\'y conjecture are deduced as consequences of a simple idea that I call the averaging trick. This idea is implicitly used in the literature, but no attempt was made to formalize the proof…
Most slowly synchronizing automata over binary alphabets are circular, i.e., containing a letter permuting the states in a single cycle, and their set of synchronizing words has maximal state complexity, which also implies complete…
In 1965 Erd\H os conjectured that for all $k\ge2$, $s\ge1$ and $n\ge k(s+1)$, an $n$-vertex $k$-uniform hypergraph $\F$ with $\nu(\F)=s$ cannot have more than \newline $\max\{\binom{sk+k-1}k,\;\binom nk-\binom{n-s}k\}$ edges. It took almost…
In 2007, Dmytrenko, Lazebnik and Williford posed two related conjectures about polynomials over finite fields. Conjecture~1 is a claim about the uniqueness of certain monomial graphs. Conjecture~2, which implies Conjecture~1, deals with…
We study a connection between synchronizing automata and its set $M$ of minimal reset words, i.e., such that no proper factor is a reset word. We first show that any synchronizing automaton having the set of minimal reset words whose set of…
One deals with r-regular bipartite graphs with 2n vertices. In a previous paper Butera, Pernici, and the author have introduced a quantity d(i), a function of the number of i-matchings, and conjectured that as n goes to infinity the…
In the Shortest Common Superstring problem, one needs to find the shortest superstring for a set of strings. This problem is APX-hard, and many approximation algorithms were proposed, with the current best approximation factor of 2.466.…
A simple graph more often than not contains adjacent vertices with equal degrees. This in particular holds for all pairs of neighbours in regular graphs, while a lot such pairs can be expected e.g. in many random models. Is there a…
Given a fixed hypergraph $H$, let $\mbox{wsat}(n,H)$ denote the smallest number of edges in an $n$-vertex hypergraph $G$, with the property that one can sequentially add the edges missing from $G$, so that whenever an edge is added, a new…
Using combinatorial properties of incomplete sets in a free monoid we construct a series of n-state deterministic automata with zero whose shortest synchronizing word has length n^2/4+n/2-1.
We prove the Cerny conjecture for one-cluster automata with prime length cycle. Consequences are given for the hybrid Road-coloring-Cerny conjecture for digraphs with a proper cycle of prime length.
It has been known since the 60's that any complete discrete $n$-state automaton admits a reset word of length not exceeding $\alpha n^3+o(n^3)$ for some absolute constant $\alpha$. J.-E. Pin and P. Frankl proved this statement with…
Let $W^{(n)}$ be the $n$-letter word obtained by repeating a fixed word $W$, and let $R_n$ be a random $n$-letter word over the same alphabet. We show several results about the length of the longest common subsequence (LCS) between…
Let D denote an infinite alphabet -- a set that consists of infinitely many symbols. A word w = a_0 b_0 a_1 b_1 ... a_n b_n of even length over D can be viewed as a directed graph G_w whose vertices are the symbols that appear in w, and the…
Consider $ A^* $, the free monoid generated by the finite alphabet $A$ with the concatenation operation. Two words have the same commutative image when one is a permutation of the symbols of the other. The commutative closure of a set $ L…
The communication matrix for two-way deterministic finite automata (2DFA) with $n$ states is defined for an automaton over a full alphabet of all $(2n+1)^n$ possible symbols: its rows and columns are indexed by strings, and the entry $(u,…
A Wheeler automaton is a finite state automaton whose states admit a total Wheeler order, reflecting the co-lexicographic order of the strings labeling source-to-node paths. A Wheeler language is a regular language admitting an accepting…
The model of generalized automata, introduced by Eilenberg in 1974, allows representing a regular language more concisely than conventional automata by allowing edges to be labeled not only with characters, but also strings. Giammarresi and…
In 1987 Hiroshi Maehara conjectured that a graph can be represented by vectors considered adjacent when not orthogonal (a faithful orthogonal representation) in codimension the minimum degree of the graph. Without settling the conjecture,…
Graph matching, also known as network alignment, refers to finding a bijection between the vertex sets of two given graphs so as to maximally align their edges. This fundamental computational problem arises frequently in multiple fields…