University of IsfahanTransactions on Combinatorics2251-86576320170901Common extremal graphs for three inequalities involving domination parameters192146410.22108/toc.2017.21464ENVladimirSamodivkinUniversity of Architecture, Civil Engineering and Geodesy (UACEG)Journal Article20170126Let $\delta (G)$, $\Delta (G)$ and $\gamma(G)$ be the minimum degree, maximum degree and domination number of a graph $G=(V(G), E(G))$, respectively. A partition of $V(G)$, all of whose classes are dominating sets in $G$, is called a domatic partition of $G$. The maximum number of classes of a domatic partition of $G$ is called the domatic number of $G$, denoted $d(G)$. It is well known that $d(G) \leq \delta(G) + 1$, $d(G)\gamma(G) \leq |V(G)|$ \cite{ch}, and $|V(G)| \leq (\Delta(G)+1)\gamma(G)$ \cite{berge}. In this paper, we investigate the graphs $G$ for which all the above inequalities become simultaneously equalities.https://toc.ui.ac.ir/article_21464_e634e304e912f76a101c385fa80076eb.pdfUniversity of IsfahanTransactions on Combinatorics2251-86576320170901On the hilbert series of binomial edge ideals of generalized trees11182146310.22108/toc.2017.21463ENMahdisSaeediAmirkabir University of TechnologyFarhadRahmatiAmirkabir University of TechnologyJournal Article20160620In this paper we introduce the concept of generalized trees and compute the Hilbert series of their binomial edge ideals.https://toc.ui.ac.ir/article_21463_658ac2a187cd8b5573536a652113719e.pdfUniversity of IsfahanTransactions on Combinatorics2251-86576320170901Binary sequence/array pairs via diference set pairs: A recursive approach19362146610.22108/toc.2017.21466ENK. T.ArasuWright State UniversityAnikaGoyalDept. of Computer Engg., YMCA University of Science And Technology, Faridabad, HR 121006, IndiaAbhishekPuriDept. of Computer Engg., YMCA University of Science And Technology, Faridabad, HR 121006, IndiaJournal Article20150216Binary array pairs with optimal/ideal correlation values and their algebraic counterparts "difference set pairs" (DSPs) in abelian groups are studied. In addition to generalizing known 1-dimensional (sequences) examples, we provide four new recursive constructions, unifying previously obtained ones. Any further advancements in the construction of binary sequences/arrays with optimal/ideal correlation values (equivalently cyclic/abelian difference sets) would give rise to richer classes of DSPs (and hence binary perfect array pairs). Discrete signals arising from DSPs find applications in cryptography, CDMA systems, radar and wireless communications.https://toc.ui.ac.ir/article_21466_7872b534dc27cd9c3fa2ab7a0cb15ba8.pdfUniversity of IsfahanTransactions on Combinatorics2251-86576320170901A class of Ramsey-extremal hypergraphs37432146810.22108/toc.2017.21468ENBrendan D.McKayAustralian National UniversityJournal Article20161204In 1991, McKay and Radziszowski proved that, however each $3$-subset of a $13$-set is assigned one of two colours, there is some $4$-subset whose four $3$-subsets have the same colour. More than 25 years later, this remains the only non-trivial classical Ramsey number known for hypergraphs. In this article, we find all the extremal colourings of the $3$-subsets of a 12-set and list some of their properties. We also provide an answer to a question of Dudek, La Fleur, Mubayi and R\"odl about the size-Ramsey numbers of hypergraphs.https://toc.ui.ac.ir/article_21468_114fdfc65b03414f07a821ae0e7d6b38.pdfUniversity of IsfahanTransactions on Combinatorics2251-86576320170901Distance in cayley graphs on permutation groups generated by $k$ $m$-Cycles45592147310.22108/toc.2017.21473ENZohrehMostaghimIran University of Science and TechnologyMohammad HosseinGhaffariIran University of Science and TechnologyJournal Article20161211In this paper, we extend upon the results of B. Suceav{\u{a}} and R. Stong [Amer. Math. Monthly, 110 (2003) 162--162], which they computed the minimum number of 3-cycles needed to generate an even permutation.<br />Let $\Omega^n_{k,m}$ be the set of all permutations of the form $c_1 c_2 \cdots c_k$ where $c_i$'s are arbitrary $m$-cycles in $S_n$. Suppose that $\Gamma^n_{k,m}$ be the Cayley graph on subgroup of $S_n$ generated by all permutations in $\Omega^n_{k,m}$. We find a shortest path joining identity and any vertex of $\Gamma^n_{k,m}$, for arbitrary natural number $k$, and $m=2 , 3, 4$. Also, we calculate the diameter of these Cayley graphs. As an application, we present an algorithm for finding a short expression of a permutation as products of given permutations.<br />https://toc.ui.ac.ir/article_21473_2e07c04c5fad360f2c8b9fc03265c648.pdf