Small cycles, generalized prisms and Hamiltonian cycles in the Bubble-sort graph

The Bubble-sort graph $BS_n,\,n\geqslant 2$, is a Cayley graph over the symmetric group $Sym_n$ generated by transpositions from the set $\{(1 2), (2 3),\ldots, (n-1 n)\}$. It is a bipartite graph containing all even cycles of length $\ell$, where $4\leqslant \ell\leqslant n!$. We give an explicit combinatorial characterization of all its $4$- and $6$-cycles. Based on this characterization, we define generalized prisms in $BS_n,\,n\geqslant 5$, and present a new approach to construct a Hamiltonian cycle based on these generalized prisms.Comment: 13 pages, 7 figure

Reconstruction of permutations distorted by single transposition errors

The reconstruction problem for permutations on $n$ elements from their erroneous patterns which are distorted by transpositions is presented in this paper. It is shown that for any $n \geq 3$ an unknown permutation is uniquely reconstructible from 4 distinct permutations at transposition distance at most one from the unknown permutation. The {\it transposition distance} between two permutations is defined as the least number of transpositions needed to transform one into the other. The proposed approach is based on the investigation of structural properties of a corresponding Cayley graph. In the case of at most two transposition errors it is shown that $\frac32(n-2)(n+1)$ erroneous patterns are required in order to reconstruct an unknown permutation. Similar results are obtained for two particular cases when permutations are distorted by given transpositions. These results confirm some bounds for regular graphs which are also presented in this paper.Comment: 5 pages, Report of paper presented at ISIT-200

An improved bound on the chromatic number of the Pancake graphs

In this paper an improved bound on the chromatic number of the Pancake graph $P_n, n\geqslant 2$, is presented. The bound is obtained using a subadditivity property of the chromatic number of the Pancake graph. We also investigate an equitable coloring of $P_n$. An equitable $(n-1)$-coloring based on efficient dominating sets is given and optimal equitable $4$-colorings are considered for small $n$. It is conjectured that the chromatic number of $P_n$ coincides with its equitable chromatic number for any $n\geqslant 2$

The Wiener Polynomial Derivatives and Other Topological Indices in Chemical Research

Wiener polynomial derivatives and some other information and topological indices are investigated with respect to their discriminating power and property correlating ability

Два случаја на несиндромска конгенитална унилатерална хипоплазија во една фамилија

Micromastia or breast hypoplasia is described as underdevelopment of a woman's mammary tissue. We present the case of a 15-year-old girl with unilateral micromastia, with familial predisposition. Ultrasound, hormonal, dysmorphic, cardiologic, genetic examinations and testing were performed. No mutation in the whole- exome sequencing was found, nor novel mutation. Some of these cases have been reported to be related to breаst cancer so further follow-up is mandatory. Therapy consists of surgical reconstruction of the affected breast. This is a rare condition and it requires a multidisciplinary approach.Микромастија или хипоплазија на дојки е опишана како недоразвиеност на ткивото на дојката кај жените. Опишавме случај на 15-годишно девојче со унилатерална микромастија со фамилијарна предиспозиција. Беа направени ехосонографски, хормонални, дисморфични, кардиолошки и генетски испитувања и тестови. Не беше пронајдена мутација на целокупниот секвенциониран ексом, ниту пак нова мутација. Некои од овие случаи се прикажани како да се поврзани со канцер на дојката и затоа се потребни понатамошни задолжителни следења. Терапијата се состои од хируршка реконструкција на афектираната дојка. Ова претставува ретка состојба, но бара мултидисциплинарен пристап