Olga Bodroža-Pantić

• PhD
• Professor (Full) at University of Novi Sad

We prove that the transfer digraph D* C,m needed for the enumeration of 2-factors in the thin cylinder TnCm(n), torus TGm(n) and Klein bottle KBm(n) (all grid graphs of the fixed width m and with m?n vertices), when m is odd, has only two components of order 2m?1 which are isomorphic. When m is even, D* C,m has [m/2] + 1 components which orders can...

In this paper, we propose an algorithm for obtaining the common transfer digraph Dm* for enumeration of 2-factors in graphs from the title, all of which have m·n vertices (m,n∈N,m≥2). The numerical data gathered for m≤18 reveal some matches for the numbers of 2-factors for different types of torus or Klein bottle. In the latter case, we conjecture...

In this paper, we prove that all but one of the components of the transfer digraph D? m needed for the enumeration of 2-factors in the rectangular, thick cylinder and Moebius strip grid graphs of the fixed width m (m ? N) are bipartite digraphs and that their orders could be expressed in term of binomial coefficients. In addition, we prove that the...

We prove that the transfer digraph ${\cal D}^*_{C,m}$ needed for the enumeration of 2-factors in the thin cylinder $TnC_{m}(n)$, torus $TG_{m}(n)$ and Klein bottle $KB_m(n)$ (all grid graphs of the fixed width $m$ and with $m \cdot n$ vertices), when $m$ is odd, has only two components of order $2^{m-1}$ which are isomorphic. When $m$ is even, ${\c...

In this paper, we prove that all but one of the components of the transfer digraph ${\cal D}^*_m$ needed for the enumeration of 2-factors in the rectangular, thick cylinder and Moebius strip grid graphs of the fixed width $m$ $(m \in N)$ are bipartite digraphs and that their orders could be expressed in term of binomial coefficients. In addition, w...

We propose an algorithm for obtaining the common transfer digraph $ D^*_m$ for enumeration of 2-factors in graphs from the title all of which with $m n$ vertices ($m, n \in N, m >1 $). The numerical data gathered for $m <19$ reveal some matchings of the numbers of 2-factors for different types of torus or Klein bottle. In latter case we conjecture...

Motivated to find the answers to some of the questions that have occurred in recent papers dealing with Hamiltonian cycles (abbreviated HCs) in some special classes of grid graphs we started the investigation of spanning unions of cycles, the so-called 2-factors, in these graphs (as a generalizations of HCs). For all the three types of graphs from...

In this series of papers, the primary goal is to enumerate Hamiltonian cycles (HC's) on the grid cylinder graphs $P_{m+1}\times C_n$, where $n$ is allowed to grow whilst $m$ is fixed. In Part~I, we studied the so-called non-contractible HC's. Here, in Part~II, we proceed further on to the contractible case. We propose two different novel characteri...

Here, in Part II, we proceeded further with the enumeration of Hamiltonian cycles (HC's) on the grid cylinder graphs of the form Pm+1?Cn, where n is allowed to grow and m is fixed. We proposed two novel characterisations of the contractible HC's. Finally, we made a conjecture concerning the dependency of the asymptotically dominant type of HC's on...

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 -...

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 tota...

In a recent paper, we have studied the enumeration of Hamiltonian cycles (abbreviated HCs) on the grid cylinder graph P m+1 × C n , where m grows while n is fixed. In this sequel, we study a much harder problem of enumerating HCs on the same graph only this time letting n grow while m is fixed. We propose a characterization for non-contractible HCs...

19 We continue our research in the enumeration of Hamiltonian cycles (HC) 20 on thin cylinder grid graphs C m × P n+1 by studying a triangular variant of 21 the problem. There are two types of HCs, distinguished by whether they 22 wrap around the cylinder. Using two characterizations of these HCs, we 23 prove that, for fixed m, the number of HCs of...

In the studies that have been devoted to the protein folding problem, which is one of the great unsolved problems of science, some specific graphs, like the so-called triangular grid graphs, have been used as a simplified lattice model. Generation and enumeration of Hamiltonian paths and Hamiltonian circuits (compact conformations of a chain) are n...

Graph Theory International audience We study the enumeration of Hamiltonian cycles on the thin grid cylinder graph $C_m \times P_{n+1}$. We distinguish two types of Hamiltonian cycles, and denote their numbers $h_m^A(n)$ and $h_m^B(n)$. For fixed $m$, both of them satisfy linear homogeneous recurrence relations with constant coefficients, and we de...

We study the enumeration of Hamiltonian cycles on the thin grid cylinder graph Cm x Pn+1. We distinguish two types of Hamiltonian cycles depending on their contractibility (as Jordan curves) and denote their numbers hmnc (n) and hmc (n). For fixed m, both of them satisfy linear homogeneous recurrence relations with constant coefficients. We derive...

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 ideal...

In this paper the didactically-methodological procedure named the MTE-model of mathematics teaching (Motivation test-Teaching-Examination test) is suggested and recommended when the teacher has subsequent lessons. This model is presented in detail through the processing of a nonstandard theme - the theme of decomposition of planes. Its efficiency h...

Intuitively obvious theorems which are hard to prove are nothing new in topology. The most celebrated case is certainly the Jordan curve theorem. For pedagogical reasons elementary proofs of such theorems never become obsolete. During their work with students of mathematics the following problem has forced itself on the authors of the paper: to pro...

The concept of ASC (Algebraic structure count) is introduced into theoretical organic chemistry by Wilcox as the difference between the number of so-called “even” and “odd” Kekulé structures of a conjugated molecule. Precisely, algebraic structure count (ASC-value) of the bipartite graph G corresponding to the skeleton of a conjugated hydrocarbon i...

The algebraic structure count of a graph G can be defined by ASC{G}=|detA| where A is the adjacency matrix of G. In chemistry, the thermodynamic stability of a hydrocarbon is related to the ASC-value for the graph which represents its skeleton. In the case of benzenoid graphs (connected, bipartite, plane graphs which have the property that every fa...

The well known game of Hex uses the following corollary of the Jordan Curve Theorem, so-called theta-Curve Theorem: An open Jordan curve with its endpoints on a closed Jordan curve K, but otherwise located in the bounded part, divides the closure of the bounded part into two parts. Some students in a honor class, who have not been previously expose...

The concept of ASC (algebraic structure count) is introduced into theoretical organic chemistry by Wilcox as the difference between the number of so-called ASC{ G} = Ö{|detA|}ASC\{ G\} = \sqrt {|\det A|} where A is the adjacency matrix of G. In the case of bipartite planar graphs containing only circuits of the length of the form 4s+2 (s=1,2,...)...

The subject of the study are the elementary excitations in the rectangular quantum wire with ”free” surfaces. We shall study a completely general quadratic harmonic Hamiltonian where the quasiparticle kinematics (Fermi or Bose) is irrelevant due to the use of the single-particle wave-function. We offer a ”nonstandard” approach to the solution of the...

The algebraic structure count (A) of a class of conjugated polymers is determined, Both explicit combinatorial formulas and recurrence relations for A are obtained. The conjugated polymers considered consist of arbitrary monomer units whose all Kekulé structures are of equal parity, which are joined via four-membered (cyclobutadiene) rings. We find...

The algebraic structure count (A) of a class of conjugated polymers is determined. Both explicit combinatorial formulas and recurrence relations for A are obtained. The conjugated polymers considered consist of arbitrary monomer units whose all Kekule structures are of equal parity, which are joined via four-membered (cyclobutadiene) rings. We find...

Let f m (n) and h m (n) denote the number of two-factors and the number of connected two-factors (Hamiltonian cycles) respectively in a (m-1)×(n-1) grid, i.e. in the labelled graph P m ×P n . We show that for each fixed m (m>1) the sequences f m =(f m (2),f m (3),...) and h m =(h m (2),h m (3),...) satisfy difference equations (linear, homogeneous,...

An algorithm which generates and enumerates Hamiltonian cycles in P m ×P n (rectangular lattice graph) is offered. It was implemented in PASCAL.

A recurrence relation for the number of 2-factors of the Cartesian product P 5 ×P n is derived. By solving the recurrence relation an explicit formula for the number f(n) of 2-factors of P 5 ×P n is obtained.

A new polynomial expression is obtained for the number of Kekulé structures (K numbers) of an arbitrary unbranched benzenoid chain composed from n linearly condensed segments containing x 1 ,x 2 ,···,x n hexagons respectively.

An explicit combinatorial formula for the number of Kekul structures of a hexagon-shaped benzencid system is deduced. Thus, the validity of the previously proposed, but hitherto unproved formulas of Everett (from 195'1), Woodger (from 1951), and Cyvin from 1986) is confirmed. The proof is based on the application of the John-Sachs theorem.