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A μ-way (v, k, t) trade is a pair T=(X,{T1,T2,...,Tμ}) such that for eacht-subset of v-set X the number of blocks containing this t-subset is the same in each Ti(1≤i≤μ). In the other words for each 1≤i

Let G = (V (G);E(G)) be a graph, γt(G). Let ooir(G) be the total domination and OO-irredundance number of G, respectively. A total dominating set S of G is called a total perfect code if every vertex in V (G) is adjacent to exactly one vertex of S. In this paper, we show that if G has a total perfect code, then γt(G) = ooir(G). As a consequence, we determine the value of ooir(G) for some classes of graphs. Finally, we prove some new bounds for the total subdivision number. [ABSTRACT FROM AUTHOR]

Two Latin squares of order n are orthogonal if in their superposition, each of the n2 ordered pairs of symbols occurs exactly once. Colbourn, Zhang and Zhu, in a series of papers, determined the integers r for which there exist a pair of Latin squares of order n having exactly r different ordered pairs in their superposition. Dukes and Howell defined the same problem for Latin squares of different orders n and n+k. They obtained a non-trivial lower bound for r and solved the problem for k ≥2n/3 . Here for k < 2n/3 , some constructions are shown to realize many values of r and for small cases (3 ≤ n ≤ 6), the problem has been solved. [ABSTRACT FROM AUTHOR]

We continue the study of restrained double Roman domination in graphs. For a graph G = (V (G) , E (G)) , a double Roman dominating function f is called a restrained double Roman dominating function (RDRD function) if the subgraph induced by { v ∈ V (G) ∣ f (v) = 0 } has no isolated vertices. The restrained double Roman domination number (RDRD number) γ r d R (G) is the minimum weight ∑ v ∈ V (G) f (v) taken over all RDRD functions of G. We first prove that the problem of computing γ r d R is NP-hard even for planar graphs, but it is solvable in linear time when restricted to bounded clique-width graphs such as trees, cographs and distance-hereditary graphs. Relationships between γ r d R and some well-known parameters such as restrained domination number γ r , domination number γ or restrained Roman domination number γ r R are investigated in this paper by bounding γ r d R from below and above involving γ r , γ or γ r R for general graphs, respectively. We prove that γ r d R (T) ≥ n + 2 for any tree T ≠ K 1 , n − 1 of order n ≥ 2 and characterize the family of all trees attaining the lower bound. The characterization of graphs with small RDRD numbers is given in this paper. [ABSTRACT FROM AUTHOR]

A μ-way (v, k, t) trade of volume m consists of μ disjoint collections T1, T2, . . . Tμ, each of m blocks, such that for every t-subset of v-set V the number of blocks containing this t-subset is the same in each Ti (1 ≤ i ≤ μ). In other words any pair of collections {Ti, Tj}, 1 ≤ i < j ≤ μ is a (v, k, t) trade of volume m. In this paper we investigate the existence of ≥-way (v, k, t) trades and prove the existence of. (i) 3-way (v, k, 1) trades Steiner trades) of each volume m,m ≥ 2. (ii) 3-way (v, k, 2) trades of each volume m,m ≥ 6 except possibly m = 7. We establish the non-existence of 3-way (v, 3, 2) trade of volume 7. It is shown that the volume of a 3-way (v, k, 2) Steiner trade is at least 2k for k ≥ 4. Also the spectrum of 3-way (v, k, 2) Steiner trades for k = 3 and 4 are specified. [ABSTRACT FROM AUTHOR]

A zero forcing set is a new concept in Graph Theory which was introduced in recent years. In this paper, we investigate the relationship between zero forcing sets and algebraic hyperstructures. To this end, we present some new definitions by considering a zero forcing process on a graph G. These definitions help us analyze the zero forcing process better and construct various hypergroups and join spaces on the vertex set of graph G. Finally, we give some examples to clarify these hyperstructures. [ABSTRACT FROM AUTHOR]

In this paper, we show that for all v ≡ 1 (mod 3), there exists a super-simple ( v , 4 , 2 ) directed design. Also, we show that for these parameters there exists a super-simple ( v , 4 , 2 ) directed design whose each defining set has at least a half of the blocks. [ABSTRACT FROM AUTHOR]