Given a set D = {d1, d2, . . .} of positive integers, one defines a distance graph with the set of integers Z as the vertex set and xy an edge iff |x−y| ∈ D. We approach distance graphs by using p-adic methods. This allows us to give general bounds on the chromatic number that depend on the divisibility properties of the numbers di. Furthermore, the chromatic number is determined for large clas...

This paper develops a theory for characterisation of DNA sequences based on their measure representation. The measures are shown to be random cascades generated by an infinitely divisible distribution. This probability distribution is uniquely determined by the exponent function in the multifractal theory of random cascades. Curve fitting to a large number of complete genomes of bacteria indica...

Divisibility monoids are a natural lattice-theoretical generalization of Mazurkiewicz trace monoids, namely monoids in which the distributivity of the involved divisibility lattices is kept as an hypothesis, but the relations between the generators are not supposed to necessarily be commutations. Here, we show that every divisibility monoid admits an explicit finite transducer which allows to c...

A graph G is said to be 2-divisible if for all (nonempty) induced subgraphs H of G, V (H) can be partitioned into two sets A,B such that ω(A) < ω(H) and ω(B) < ω(H). A graph G is said to be perfectly divisible if for all induced subgraphs H of G, V (H) can be partitioned into two sets A,B such that H[A] is perfect and ω(B) < ω(H). We prove that if a graph is (P5, C5)-free, then it is 2-divisibl...

For a fixed graph H with t vertices, an H-factor of a graph G with n vertices, where t divides n, is a collection of vertex disjoint (not necessarily induced) copies of H in G covering all vertices of G. We prove that for a fixed tree T on t vertices and > 0, the random graph Gn,p, with n a multiple of t, with high probability contains a family of edge-disjoint T -factors covering all but an -f...

We address two sets of long-standing open questions in linear algebra and probability theory, from a computational complexity perspective: stochastic matrix divisibility, and divisibility and decomposability of probability distributions. We prove that finite divisibility of stochastic matrices is an NP-complete problem, and extend this result to nonnegative matrices, and completely-positive tra...

We continue the study of a recently introduced model of communication complexity with advice, focusing on the power of advice in the context of equality of bitstrings and divisibility of natural numbers. First, we establish that the equality problem admits a protocol of polylogarithmic communication, provided a laconic advice of just one bit. For the divisibility problem, we design a protocol w...