Bojana Pantić’s research while affiliated with University of Novi Sad and other places

This short note deals with the so-called {\em Sock Matching Problem} which appeared in [S. Gilliand, C. Johnson, S. Rush \& D. Wood, The sock matching problem, {\em Involve}, 7 (5) (2014), 691--697.]. Let us denote by $B_{n,k}$ the number of all the sequences $a_1, \ldots, a_{2n}$ of nonnegative integers with $a_1 = 1$, $a_{2n}=0$ and $\mid a_i -a_{i+1}\mid \; = 1$ containing at least one number k $(1 \leq k \leq n)$. The value $a_i$ can be interpreted as the number of unmatched socks being present after drawing the first i socks randomly out of the pile which initially contained n pairs of socks. Here, establishing a link between this problem and with both some old and some new results, related to the number of restricted Dyck paths, we prove that the probability for k unmatched socks to appear (in the very process of drawing one sock at a time) approaches 1 as the number of socks becomes large enough. Furthermore, we obtain a few valid forms of the sock matching theorem, thereby at the same time correcting the omissions made in the above mentioned paper.

For all odd values of m, we prove that the sequence of the numbers of near-perfect matchings on Cm x P2n+1 cylinder with a vacancy on the boundary obeys the same recurrence relation as the sequence of the numbers of perfect matchings on Cm x P2n. Further more, we prove that for all odd values of m denominator of the generating function for the total number of the near-perfect matchings on Cm x P2n+1 graph is always the square of denominator of generating function for the sequence of the numbers of perfect matchings on Cm x P2n graph, as recently conjectured by Perepechko.

In contrast to the abundance of "direct" Ramsey results for classes of finite structures (such as finite ordered graphs, finite ordered metric spaces and finite posets with a linear extension), in only a handful of cases we have a meaningful dual Ramsey result. In this paper we prove a dual Ramsey theorem for finite ordered oriented graphs. Instead of embeddings, which are crucial for "direct" Ramsey results, we consider a special class of surjective homomorphisms between finite ordered oriented graphs. Since the setting we are interested in involves both structures and morphisms, all our results are spelled out using the reinterpretation of the (dual) Ramsey property in the language of category theory.

In contrast to the abundance of "direct" Ramsey results for classes of finite structures (such as finite ordered graphs, finite ordered metric spaces and finite posets with a linear extension), in only a handful of cases we have a meaningful dual Ramsey result. In this paper we prove a dual Ramsey theorem for finite ordered oriented graphs. Instead of embeddings, which are crucial for "direct" Ramsey results, we consider a special class of surjective homomorphisms between finite ordered oriented graphs. Since the setting we are interested in involves both structures and morphisms, all our results are spelled out using the reinterpretation of the (dual) Ramsey property in the language of category theory.

In polymer science, Hamiltonian paths and Hamiltonian circuits can serve as excellent simple models for dense packed globular proteins. Generation and enumeration of Hamiltonian paths and Hamiltonian circuits (compact conformations of a chain) are needed to investigate thermodynamics of protein folding. Hamiltonian circuits are a mathematical idealization of polymer melts, too. The number of Hamiltonian cycles on a graph corresponds to the entropy of a polymer system. We present new characterizations of the Hamiltonian cycles in a labeled rectangular grid graph P m ×P n and in a labeled thin cylinder grid graph C m ×P n . We proved that for any fixed m, the numbers of Hamiltonian cycles in these grid graphs, as sequences with counter n, are determined by linear recurrences. The computational method outlined here for finding these difference equations together with the initial terms of the sequences has been implemented. The generating functions of the sequences are given explicitly for some values of m. The obtained data are consistent with data obtained in the works by Kloczkowski and Jernigan, and Schmalz et al.