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... more
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 [Formula: see text]. These definitions help us analyze the zero forcing process better and construct various hypergroups and join spaces on the vertex set of graph [Formula: see text]. Finally, we give some examples to clarify these hyperstructures.
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A (v, k, t) directed trade (or simply a (v, k, t)DT) of volume s consists of two disjoint collections Tl and each containing ordered k-tuples of distinct elements of a v-set called blocks, such that the number of blocks containing any... more
A (v, k, t) directed trade (or simply a (v, k, t)DT) of volume s consists of two disjoint collections Tl and each containing ordered k-tuples of distinct elements of a v-set called blocks, such that the number of blocks containing any t-tuple of V is the same in Tl as in T2 . Our study shows that the volume of a (v, k, t)DT is at least 2Lt/ 2J and that directed trades with minimum volume and minimum foundation exist. Also it is shown that for each s 2:: 2, there exists a (v, k, 2)DT and a (v, k,3)DT each of volume s, with one exception, that is, no (v,4, 3)DT of volume three exists.
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The flower at a point x in a Steiner triple system (X; B) is the set of all triples containing x. Denote by J3F(r) the set of all integers k such that there exists a collection of three STS(2r+1) mutually intersecting in the same set of k... more
The flower at a point x in a Steiner triple system (X; B) is the set of all triples containing x. Denote by J3F(r) the set of all integers k such that there exists a collection of three STS(2r+1) mutually intersecting in the same set of k + r triples, r of them being the triples of a common flower. In this article we determine the set J3F(r) for any positive integer r = 0, 1 (mod 3) (only some cases are left undecided for r = 6, 7, 9, 24), and establish that J3F(r) = I3F(r) for r = 0, 1 (mod 3) where I3F(r) = {0, 1,..., 2r(r-1)/3-8, 2r(r-1)/3-6, 2r(r-1)/3}.
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A 3-way (v,k,t) trade T of volume m consists of three pairwise disjoint collections T_1, T_2 and T_3, each of m blocks of size k, such that for every t-subset of v-set V, the number of blocks containing this t-subset is the same in each... more
A 3-way (v,k,t) trade T of volume m consists of three pairwise disjoint collections T_1, T_2 and T_3, each of m blocks of size k, such that for every t-subset of v-set V, the number of blocks containing this t-subset is the same in each T_i for 1≤ i≤ 3. If any t-subset of found(T) occurs at most once in each T_i for 1≤ i≤ 3, then T is called 3-way (v,k,t) Steiner trade. We attempt to complete the spectrum S_3s(v,k), the set of all possible volume sizes, for 3-way (v,k,2) Steiner trades, by applying some block designs, such as BIBDs, RBs, GDDs, RGDDs, and r× s packing grid blocks. Previously, we obtained some results about the existence some 3-way (v,k,2) Steiner trades. In particular, we proved that there exists a 3-way (v,k,2) Steiner trade of volume m when 12(k-1)≤ m for 15≤ k (Rashidi and Soltankhah, 2016). Now, we show that the claim is correct also for k≤ 14.
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... more
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.
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In this paper we investigate the spectrum of super-simple 2-$(v,5,1)$ directed designs (or simply super-simple 2-$(v,5,1)$DDs) and also the size of their smallest defining sets. We show that for all $v\equiv1,5\ ({\rm mod}\ 10)$ except... more
In this paper we investigate the spectrum of super-simple 2-$(v,5,1)$ directed designs (or simply super-simple 2-$(v,5,1)$DDs) and also the size of their smallest defining sets. We show that for all $v\equiv1,5\ ({\rm mod}\ 10)$ except $v=5,15$ there exists a super-simple $(v,5,1)DD$. Also for these parameters, except possibly $v=11,91$, there exists a super-simple 2-$(v,5,1)$DD whose smallest defining sets have at least a half of the blocks.
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In a given graph $G$, a set $S$ of vertices with an assignment of colors is a {\sf defining set of the vertex coloring of $G$}, if there exists a unique extension of the colors of $S$ to a $\Cchi(G)$-coloring of the vertices of $G$. A... more
In a given graph $G$, a set $S$ of vertices with an assignment of colors is a {\sf defining set of the vertex coloring of $G$}, if there exists a unique extension of the colors of $S$ to a $\Cchi(G)$-coloring of the vertices of $G$. A defining set with minimum cardinality is called a {\sf smallest defining set} (of vertex coloring) and its cardinality, the {\sf defining number}, is denoted by $d(G, \Cchi)$. Let $ d(n, r, \Cchi = k)$ be the smallest defining number of all $r$-regular $k$-chromatic graphs with $n$ vertices. Mahmoodian et. al \cite{rkgraph} proved that, for a given $k$ and for all $n \geq 3k$, if $r \geq 2(k-1)$ then $d(n, r, \Cchi = k)=k-1$. In this paper we show that for a given $k$ and for all $n < 3k$ and $r\geq 2(k-1)$, $d(n, r, \Cchi=k)=k-1$.
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A set S of vertices in a graph G(V, E) is called a total dominating set if every vertex v ∈ V is adjacent to an element of S. A set S of vertices in a graph G(V, E) is called a total restrained dominating set if every vertex v ∈ V is... more
A set S of vertices in a graph G(V, E) is called a total dominating set if every vertex v ∈ V is adjacent to an element of S. A set S of vertices in a graph G(V, E) is called a total restrained dominating set if every vertex v ∈ V is adjacent to an element of S and every vertex of V − S is adjacent to a vertex in V − S. The total domination number of a graph G denoted by γt(G) is the minimum cardinality of a total dominating set in G. Respectively the total restrained domination number of a graph G denoted by γtr(G) is the minimum cardinality of a total restrained dominating set in G. Here we investigate the problem of total domination numbers and total restrained domination numbers of some grid graphs (cartesian products of two paths Pn and Pm). And we determine the total domination numbers of Pn,n, P2n,2n+2, P2n,4n−1, and P2n,m for each n and m ≡ 2n (mod 2n + 1). Also we determine the total domination numbers of P8,n. We then show that for these grid graphs the total restrained do...
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The concept of defining set has been studied in block designs and, under the name critical sets, in Latin squares and Room squares. Here we study defining sets for directed designs. A t-(v, k, λ) directed design (DD) is a pair (V, B),... more
The concept of defining set has been studied in block designs and, under the name critical sets, in Latin squares and Room squares. Here we study defining sets for directed designs. A t-(v, k, λ) directed design (DD) is a pair (V, B), where V is a v-set and B is a collection of ordered blocks (or k-tuples of V), for which each t-tuple of V appears in precisely λ blocks. A set of blocks which is a subset of a unique t-(v, k, λ)DD is said to be a defining set of the directed design. As in the case of block designs, finding defining sets seems to be a difficult problem. In this note we introduce some lower bounds for the number of blocks in smallest defining sets in directed designs, determine the precise number of blocks in smallest defining sets for some directed designs with small parameters and point out an open problem relating to the number of blocks needed to define a directed design as compared with the number needed to define its underlying undirected design.
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A $2-(v,k,\lambda)$ directed design (or simply a $2-(v,k,\lambda)DD$) is super-simple if its underlying $2-(v,k,2\lambda)BIBD$ is super-simple, that is, any two blocks of the $BIBD$ intersect in at most two points. A $2-(v,k,\lambda)DD$... more
A $2-(v,k,\lambda)$ directed design (or simply a $2-(v,k,\lambda)DD$) is super-simple if its underlying $2-(v,k,2\lambda)BIBD$ is super-simple, that is, any two blocks of the $BIBD$ intersect in at most two points. A $2-(v,k,\lambda)DD$ is simple if its underlying $2-(v,k,2\lambda)BIBD$ is simple, that is, it has no repeated blocks. A set of blocks which is a subset of a unique $2-(v,k,\lambda)DD$ is said to be a defining set of the directed design. A smallest defining set, is a defining set which has smallest cardinality. In this paper simultaneously we show that the necessary and sufficient condition for the existence of a super-simple $2-(v,4,1)DD$ is $v\equiv1\ ({\rm mod}\ 3)$ and for these values except $v=7$, there exists a super-simple $2-(v,4,1)DD$ whose smallest defining sets have at least a half of the blocks. And also for all $\epsilon > 0$ there exists $v_0(\epsilon)$ such that for all admissible $v>v_0$ there exists a $2-(v,4,1)DD$ whose smallest defining sets hav...
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A set S of vertices in a graph G(V, E) is called a total dominating set if every vertex v ∈ V is adjacent to an element of S. The total domination number of a graph G denoted by γt(G) is the minimum cardinality of a total dominating set... more
A set S of vertices in a graph G(V, E) is called a total dominating set if every vertex v ∈ V is adjacent to an element of S. The total domination number of a graph G denoted by γt(G) is the minimum cardinality of a total dominating set in G. Total domination subdivision number denoted by sdγt is the minimum number of edges that must be subdivided to increase the total domination number. Here we investigate the problem of total domination subdivision numbers of grid graphs Pm,n and determine the total domination subdivision numbers of grid graphs Pm,n for m =2 ,3 and 4, and n ≥ 2. Also Haynes et al. [4] showed that 1 ≤ sdγt(Pm,n) ≤ 4 for any grid graph Pm,n. We improve this bound and prove that sdγt (Pm,n) ≤ 3.
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We continue the study of restrained double Roman domination in graphs. For a graph G =
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A (v, k, t) trade can be used to construct new designs with various support sizes from a given t-design. H.L. Hwang (1986) showed the existence of (v, k, t) trades of volume 2t and the non-existence of trades of volumes less than 2t or of... more
A (v, k, t) trade can be used to construct new designs with various support sizes from a given t-design. H.L. Hwang (1986) showed the existence of (v, k, t) trades of volume 2t and the non-existence of trades of volumes less than 2t or of volume 2t + 1. In thIs paper, first we show that there exist (v, k, t) trades of volumes 2t + 2t1 (t ~ 1), 2t + 2t1 + 2t 2 (t ~ 2), 2t + 2t 1 + 2t 2 + 2t 3 (t ~ 3), and 2t+l. Then we prove that, given integers v > k > t ~ 1, there does not exist a (v, k, t) trade of volume s, where 2t < s < 2t + 2t-l.
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Mahmoodian and Soltankhah $\cite{MMS}$ conjectured that there does not exist any $t$-$(v,k)$ trade of volume $s_{i}< s <s_{i+1}$, where $s_{i}=2^{t+1}-2^{t-i}, i=0,1,..., t-1$. Also they showed that the conjecture is true for $i=0$.... more
Mahmoodian and Soltankhah $\cite{MMS}$ conjectured that there does not exist any $t$-$(v,k)$ trade of volume $s_{i}< s <s_{i+1}$, where $s_{i}=2^{t+1}-2^{t-i}, i=0,1,..., t-1$. Also they showed that the conjecture is true for $i=0$. In this paper we prove the correctness of this conjecture for Steiner trades.
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A d-biclique cover of a graph G is a collection of bicliques of G such that each edge of G is in at least d of the bicliques. The number of bicliques in a minimum d-biclique cover of G is called the d-biclique covering number of G and is... more
A d-biclique cover of a graph G is a collection of bicliques of G such that each edge of G is in at least d of the bicliques. The number of bicliques in a minimum d-biclique cover of G is called the d-biclique covering number of G and is denoted by ${bc}_d(G)$. In this paper, we present an upper bound for the d- biclique covering number of the lexicographic product of graphs. Also, we introduce some bounds of this parameter for some graph constructions and obtain the exact value of the d-biclique covering number of some graphs.
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Two Latin squares of order n are orthogonal if in their superposition, each of the n 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... more
Two Latin squares of order n are orthogonal if in their superposition, each of the n 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.
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Let D(u,k) be the maximum number m such that there exist m STS(3u)s (S,B1),..., (S,Bm) such that for each i ≠ j, Bi ∩ Bj = A, |A| = u + k, where u of the common triples form a parallel class. In this paper, we determine the number... more
Let D(u,k) be the maximum number m such that there exist m STS(3u)s (S,B1),..., (S,Bm) such that for each i ≠ j, Bi ∩ Bj = A, |A| = u + k, where u of the common triples form a parallel class. In this paper, we determine the number D(2n+1,0) for each n ≡ 0,1 (mod 3).
A t-(v, k,;\) directed design (or simply a t-(v, k, ;\)DD) is a pair (V, B), where V is a v-set and B is a collection of (transitively) ordered k-tuples of distinct elements of V, such that every ordered t-tuple of distinct elements of V... more
A t-(v, k,;\) directed design (or simply a t-(v, k, ;\)DD) is a pair (V, B), where V is a v-set and B is a collection of (transitively) ordered k-tuples of distinct elements of V, such that every ordered t-tuple of distinct elements of V belongs to exactly). elements of B. (We say that at-tuple belongs to a k-tuple, if its components are contained in that k-tuple as a set, and they appear with the same order). In this paper with a linear algebraic approach, we study the t-tuple inclusion matrices Dr k' which sheds light to the existence problem for directed designs. A~ong the results, we find the rank of this matrix in the case of 0 ~ t ~ 4. Also in the case of 0 ~ t ~ 3 , we introduce a semi-triangular basis for the null space of Df,t+l' We prove that when 0 :::; t :::; 4 , the obvious necessary conditions for the existence of t-( v, k, ;\) signed directed designs, are also sufficient. Finally we find a semi-triangular basis for the null space of D HI
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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... more
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 µ-way (v,k,t) trades and also we 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.
A u-way (v; k; t) trade is a pair T = (X; T_1; T2,...,T_u) such that for each t-subset of v-set X the number of blocks containing this t-subset is the same in each Ti (1 2 and t = 2.
A set S of vertices in a graph G(V, E) is called a dominating set if every vertex v ∈ V is either an element of S or is adjacent to an element of S. A set S of vertices in a graph G(V, E) is called a total dominating set if every vertex v... more
A set S of vertices in a graph G(V, E) is called a dominating set if every vertex v ∈ V is either an element of S or is adjacent to an element of S. A set S of vertices in a graph G(V, E) is called a total dominating set if every vertex v ∈ V is adjacent to an element of S. The domination number of a graph G denoted by γ(G) is the minimum cardinality of a dominating set in G. Respectively the total domination number of a graph G denoted by γt(G) is the minimum cardinality of a total dominating set in G. An upper bound for γt(G) which has been achieved by Cockayne and et al. in [1] is: for any graph G with no isolated vertex and maximum degree Δ(G) and n vertices, γt(G) ≤ n − Δ(G )+1 . Here we characterize bipartite graphs and trees which achieve this upper bound. Further we present some another upper and lower bounds for γt(G). Also, for circular complete graphs, we determine the value of γt(G).
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A 3-way $(v,k,t)$ trade $T$ of volume $m$ consists of three pairwise disjoint collections $T_1$, $T_2$ and $T_3$, each of $m$ blocks of size $k$, such that for every $t$-subset of $v$-set $V$, the number of blocks containing this... more
A 3-way $(v,k,t)$ trade $T$ of volume $m$ consists of three pairwise disjoint collections $T_1$, $T_2$ and $T_3$, each of $m$ blocks of size $k$, such that for every $t$-subset of $v$-set $V$, the number of blocks containing this $t$-subset is the same in each $T_i$ for $1\leq i\leq 3$. If any $t$-subset of found($T$) occurs at most once in each $T_i$ for $1\leq i\leq 3$, then $T$ is called 3-way $(v,k,t)$ Steiner trade. We attempt to complete the spectrum $S_{3s}(v,k)$, the set of all possible volume sizes, for 3-way $(v,k,2)$ Steiner trades, by applying some block designs, such as BIBDs, RBs, GDDs, RGDDs, and $r\times s$ packing grid blocks. Previously, we obtained some results about the existence some 3-way $(v,k,2)$ Steiner trades. In particular, we proved that there exists a 3-way $(v,k,2)$ Steiner trade of volume $m$ when $12(k-1)\leq m$ for $15\leq k$ (Rashidi and Soltankhah, 2016). Now, we show that the claim is correct also for $k\leq 14$.
A $\mu$-way $(v, k, t)$ trade $T = \{T_{1} , T_{2}, . . ., T_{\mu} \}$ of volume $m$ consists of $\mu$ disjoint collections $T_{1}, T_{2}, \ldots, T_{\mu}$, each of $m$ blocks of size $k$, such that for every $t$-subset of $v$-set $V$ the... more
A $\mu$-way $(v, k, t)$ trade $T = \{T_{1} , T_{2}, . . ., T_{\mu} \}$ of volume $m$ consists of $\mu$ disjoint collections $T_{1}, T_{2}, \ldots, T_{\mu}$, each of $m$ blocks of size $k$, such that for every $t$-subset of $v$-set $V$ the number of blocks containing this $t$-subset is the same in each $T_{i}$ (for $1 \leq i \leq \mu$). A $\mu$-way $(v, k, t)$ trade is called $\mu$-way $(v, k, t)$ Steiner trade if any $t$-subset of found$(T)$ occurs at most once in $T_{1}$ $(T_{j},\ j \geq 2)$. A $\mu$-way $(v,k,t)$ trade is called $d$-homogeneous if each element of $V$ occurs in precisely $d$ blocks of $T_{1}$ $(T_{j},~ j \geq 2)$. In this paper we characterize the $3$-way $3$-homogeneous $(v,3,2)$ Steiner trades of volume $v$. Also we show how to construct a $3$-way $d$-homogeneous $(v,3,2)$ Steiner trade for $d\in \{4,5,6\}$, except for seven small values of $v$.
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ABSTRACT In a given graph G, a set S of vertices with an assignment of colors is a defining set of the vertex coloring of G, if there exists a unique extension of the colors of S to a χ(G)-coloring of the vertices of G. A defining set... more
ABSTRACT In a given graph G, a set S of vertices with an assignment of colors is a defining set of the vertex coloring of G, if there exists a unique extension of the colors of S to a χ(G)-coloring of the vertices of G. A defining set with minimum cardinality is called a smallest defining set (of vertex coloring) and its cardinality, the defining number, is denoted by d(G,χ). We study the defining number of regular graphs. Let d(n,r,χ=k) be the smallest defining number of all r-regular k-chromatic graphs with n vertices, and f(n,k)=k-2 2(k-1)n+2+(k-2)(k-3) 2(k-1)· E. S. Mahmoodian and E. Mendelsohn determined the value of d(n,k,χ=k) for all k≤5, except for the case of (n,k)=(10,5). They showed that d(n,k,χ=k)=⌈f(n,k)⌉, for k≤5. They raised the following question: Is it true that for every k, there exists n 0 (k) such that for all n≥n 0 (k), we have d(n,k,χ=k)=⌈f(n,k)⌉? Here we determine the value of d(n,k,χ=k) for each k in some congruence classes of n. We show that the answer for the question above, in general, is negative. Also here, for k=6 and k=7 the value of d(n,k,χ=k) is determined except for one single case, and it is shown that d(10,5,χ=5)=6.
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A $\mu$-way $(v,k,t)$ $trade$ of volume $m$ consists of $\mu$ disjoint collections $T_1$, $T_2, \dots T_{\mu}$, 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... more
A $\mu$-way $(v,k,t)$ $trade$ of volume $m$ consists of $\mu$ disjoint collections $T_1$, $T_2, \dots T_{\mu}$, 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 $T_i\ (1\leq i\leq \mu)$. In other words any pair of collections $\{T_i,T_j\}$, $1\leq i<j \leq \mu$ is a $(v,k,t)$ trade of volume $m$. In this paper we investigate the existence of $\mu$-way $(v,k,t)$ trades and also we prove the existence of: (i)~3-way $(v,k,1)$ trades (Steiner trades) of each volume $m,m\geq2$. (ii) 3-way $(v,k,2)$ trades of each volume $m,m\geq6$ 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 $k\geq4$. Also the spectrum of 3-way $(v,k,2)$ Steiner trades for $k=3$ and 4 are specified.
A d-biclique cover of a graph G is a collection of bicliques of G such that each edge of G is in at least d of the bicliques. The number of bicliques in a minimum d-biclique cover of G is called the d-biclique covering number of G and is... more
A d-biclique cover of a graph G is a collection of bicliques of G such that each edge of G is in at least d of the bicliques. The number of bicliques in a minimum d-biclique cover of G is called the d-biclique covering number of G and is denoted by ${bc}_d(G)$. In this paper, we present an upper bound for the d- biclique covering number of the lexicographic product of graphs. Also, we introduce some bounds of this parameter for some graph constructions and obtain the exact value of the d-biclique covering number of some graphs.
ABSTRACT In a given graph G, a set S of vertices with an assignment of colors is a defining set of the vertex coloring of G, if there exists a unique extension of the colors of S to a χ(G)-coloring of the vertices of G. A defining set... more
ABSTRACT In a given graph G, a set S of vertices with an assignment of colors is a defining set of the vertex coloring of G, if there exists a unique extension of the colors of S to a χ(G)-coloring of the vertices of G. A defining set with minimum cardinality is called a smallest defining set (of vertex coloring) and its cardinality, the defining number, is denoted by d(G,χ). We study the defining number of regular graphs. Let d(n,r,χ=k) be the smallest defining number of all r-regular k-chromatic graphs with n vertices, and f(n,k)=k-2 2(k-1)n+2+(k-2)(k-3) 2(k-1)· E. S. Mahmoodian and E. Mendelsohn determined the value of d(n,k,χ=k) for all k≤5, except for the case of (n,k)=(10,5). They showed that d(n,k,χ=k)=⌈f(n,k)⌉, for k≤5. They raised the following question: Is it true that for every k, there exists n 0 (k) such that for all n≥n 0 (k), we have d(n,k,χ=k)=⌈f(n,k)⌉? Here we determine the value of d(n,k,χ=k) for each k in some congruence classes of n. We show that the answer for the question above, in general, is negative. Also here, for k=6 and k=7 the value of d(n,k,χ=k) is determined except for one single case, and it is shown that d(10,5,χ=5)=6.
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ABSTRACT In a given graph G, a set S of vertices with an assignment of colors is a defining set of the vertex coloring of G, if there exists a unique extension of the colors of S to a χ(G)-coloring of the vertices of G. A defining set... more
ABSTRACT In a given graph G, a set S of vertices with an assignment of colors is a defining set of the vertex coloring of G, if there exists a unique extension of the colors of S to a χ(G)-coloring of the vertices of G. A defining set with minimum cardinality is called a smallest defining set (of vertex coloring) and its cardinality, the defining number, is denoted by d(G,χ). Let d(n,r,χ=k) be the smallest defining number of all r-regular k-chromatic graphs with n vertices. E. S. Mahmoodian and E. Mendelsohn proved that for each n and each r≥4, d(n,r,χ=3)=2. They raised the following question: Is it true that for every k, there exist n 0 (k) and r 0 (k), such that for all n≥n 0 (k) and r≥r 0 (k) we have d(n,r,χ=k)=k-1? We show that the answer to this question is positive, and we prove that for a given k and for all n≥3k, if r≥2(k-1) then d(n,r,χ=k)=k-1.
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ABSTRACT A 3-way (v, k, t) trade of volume s consists of 3 disjoint collections and , each of s blocks, such that for every t-subset of v-set V, the number of blocks containing this t-subset is the same in each ( ). In this paper we prove... more
ABSTRACT A 3-way (v, k, t) trade of volume s consists of 3 disjoint collections and , each of s blocks, such that for every t-subset of v-set V, the number of blocks containing this t-subset is the same in each ( ). In this paper we prove the existence of: (i) 3-way (v, k, 1) trades (Steiner trades) of each volume . (ii) 3-way (v, k, 2) trades of each volume except . 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 . Also the spectrum of 3-way ( ) Steiner trades for and 4 are specified.