Common extremal graphs for three inequalities involving domination parameters
Vladimir
Samodivkin
University of Architecture, Civil Engineering and Geodesy (UACEG)
author
text
article
2017
eng
Let $\delta (G)$, $\Delta (G)$ 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.
Transactions on Combinatorics
University of Isfahan
2251-8657
6
v.
3
no.
2017
1
9
http://toc.ui.ac.ir/article_21464_e634e304e912f76a101c385fa80076eb.pdf
dx.doi.org/10.22108/toc.2017.21464
On the hilbert series of binomial edge ideals of generalized trees
Mahdis
Saeedi
Amirkabir University of Technology
author
Farhad
Rahmati
Amirkabir University of Technology
author
text
article
2017
eng
In this paper we introduce the concept of generalized trees and compute the Hilbert series of their binomial edge ideals.
Transactions on Combinatorics
University of Isfahan
2251-8657
6
v.
3
no.
2017
11
18
http://toc.ui.ac.ir/article_21463_658ac2a187cd8b5573536a652113719e.pdf
dx.doi.org/10.22108/toc.2017.21463
Binary sequence/array pairs via diference set pairs: A recursive approach
K. T.
Arasu
Wright State University
author
Anika
Goyal
Dept. of Computer Engg., YMCA University of Science And Technology, Faridabad, HR 121006, India
author
Abhishek
Puri
Dept. of Computer Engg., YMCA University of Science And Technology, Faridabad, HR 121006, India
author
text
article
2017
eng
Binary array pairs with optimal/ideal correlation values and their algebraic counterparts \textquotedblleft difference set pairs\textquotedblright\;(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.
Transactions on Combinatorics
University of Isfahan
2251-8657
6
v.
3
no.
2017
19
36
http://toc.ui.ac.ir/article_21466_7872b534dc27cd9c3fa2ab7a0cb15ba8.pdf
dx.doi.org/10.22108/toc.2017.21466
A class of Ramsey-extremal hypergraphs
Brendan D.
McKay
Australian National University
author
text
article
2017
eng
In 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.
Transactions on Combinatorics
University of Isfahan
2251-8657
6
v.
3
no.
2017
37
43
http://toc.ui.ac.ir/article_21468_114fdfc65b03414f07a821ae0e7d6b38.pdf
dx.doi.org/10.22108/toc.2017.21468
Distance in cayley graphs on permutation groups generated by $k$ $m$-Cycles
Zohreh
Mostaghim
Iran University of Science and Technology
author
Mohammad Hossein
Ghaffari
Iran University of Science and Technology
author
text
article
2017
eng
In 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. 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.
Transactions on Combinatorics
University of Isfahan
2251-8657
6
v.
3
no.
2017
45
59
http://toc.ui.ac.ir/article_21473_2e07c04c5fad360f2c8b9fc03265c648.pdf
dx.doi.org/10.22108/toc.2017.21473