Transactions on Combinatorics
Volume:6 Issue: 3, Sep 2017

Let δ(G)ý, ýΔ(G) and γ(G)ý ýbe the minimum degreeý, ýmaximum degree andý ýdomination number of a graph G=(V(G)ý,ýE(G))ý, ýrespectivelyý. ýA partition of V(G)ý, ýall of whose classes are dominating sets in Gý, ýis called a domatic partition of Gý. ýThe maximum number of classes ofý ýa domatic partition of G is called the domatic number of Gý, ýdenoted d(G)ý. ýIt is well known thatý ýd(G)≤δ(G)ýý1ý, ýd(G)γ(G)≤|V(G)| \cite{ch}ý, ýand |V(G)|≤(Δ(G)ýý1)γ(G) \cite{berge}ý. ýIn this paperý, ýwe investigate the graphs G for whichý ýall the above inequalities become simultaneously equalitiesý.

In this paper we introduce the concept of generalized trees and compute the Hilbert series of their binomial edge idealsý.

Binary array pairs with optimal/ideal correlation values and their algebraic counterparts \textquotedblleft difference set pairs\textquotedblright\;(DSPs) in abelian groups are studiedý. ýIn addition to generalizing known 1-dimensional (sequences) examplesý, ýwe provide four new recursive constructionsý, ýunifying previously obtained onesý. ýAny 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)ý. ýDiscrete signals arising from DSPs find applications in cryptographyý, ýCDMA systemsý, ýradar and wireless communicationsý.

In 1991ý, ýMcKay and Radziszowski proved thatý, ýhowever each 3-subset of a 13-set is assigned one of two coloursý, ýthere is some 4-subset whose four 3-subsets have the same colourý. ýMore than 25 years laterý, ýthis remains the only non-trivial classical Ramsey number known for hypergraphsý. ýIn this articleý, ýwe find all the extremal colourings of the 3-subsets of a 12-set and list some of their propertiesý. ýWe also provide an answer to a question of Dudeký, ýLa Fleurý, ýMubayi and R\"odl about the size-Ramsey numbers of hypergraphsý.

In this paperý, ýwe extend upon the results of Bý. ýSuceav{\u{a}} and Rý. ýStong [Amerý. ýMathý. ýMonthlyý, ý110 (2003) 162--162]ý, ýwhich they computed the minimum number of 3-cycles needed to generate an even permutationý.
ýLet Ωnk,m be the set of all permutations of the form c1c2⋯cký ýwhere ci's are arbitrary m-cycles in Sný. ýSuppose that Γnk,m be the Cayley graph on subgroup of Sn generated by all permutationsý ýin Ωnk,mý. ýWe find a shortest path joining identity and any vertex of Γnk,mý, ýfor arbitrary natural number ký, ýand m=2ý,ý\ý,ý3,\ý,ý4ý. ýAlsoý, ýwe calculate the diameter of these Cayley graphsý. ýAs an applicationý, ýwe present an algorithm for finding a short expression of a permutation as products of given permutationsý.