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The concept of t-(v, k) trades of block designs previously has been studied in detail. See for example A. S. Hedayat (1990) and Billington (2003). Also Latin trades have been studied in detail under various names, see A. D. Keedwell (2004) for a survey. Recently Khanban, Mahdian and Mahmoodian have extended the concept of Latin trades and introduced t-(v, k) Latin trades. Here we study the spectrum of possible volumes of these trades, S(t, k). Firstly, similarly to trades of block designs we consider (t + 2) numbers Si = 2 t+1-2(t+1)-i 0 ≤ i ≤ t + 1, as critical points and then we show that Si ∈ S(t, k), for any 0 ≤ i ≤ t + 1, and if s ∈ (si, Si+1), 0 ≤ i ≤ t, then s ∉ S(t, t + 1). As an... 

The square G2 of a graph G is a graph with the same vertex set as G in which two vertices are joined by an edge if their distance in G is at most two. For a graph G, χ[G2), which is also known as the distance two coloring number of G is studied. We study coloring the square of grids Pm□Pn, cylinders Pm□C n, and tori Cm□Cn. For each m and n we determine χ((Pm□Pn)2), χ(P m□Cn)2), and in some cases χ((C m□Cn)2) while giving sharp bounds to the latter. We show that χ((Cm□Cn)2) is at most 8 except when m -n = 3, in which case the value is 9. Moreover, we conjecture that for every m (m ≥ 5) and n (n ≥ 5), we have, 5 ≤ χ((Cm□Cn)2) ≤ 7  

For natural numbers n and k (n > 2k), a generalized Petersen graph P(n, k), is defined by vertex set {ui ui] and edge set {ui ui+1, ui ui, ui ui+k}; where i = 1, 2,..., n and subscripts are reduced modulo n. Here first, we characterize minimum vertex covers in generalized Petersen graphs. Second, we present a lower bound and some upper bounds for β(P(n, k)), the size of minimum vertex cover of P(n,k). Third, in some cases, we determine the exact values of β(P(n,k)). Our conjecture is that β(P(n, k)) < n+ [n/5], for all n and k  

A critical set in an n×n array is a set C of given entries, such that there exists a unique extension of C to an n×n Latin square and no proper subset of C has this property. For a Latin square L, scs(L) denotes the size of the smallest critical set of L, and scs(n) is the minimum of scs(L) over all Latin squares L of order n. We find an upper bound for the number of partial Latin squares of size k and prove thatn2-(e+o(1))n10/6≤maxscs(L)≤n2- π2n9/6.This improves on a result of Cavenagh (Ph.D. Thesis, The University of Queensland, 2003) and disproves one of his conjectures. Also it improves the previously known lower bound for the size of the largest critical set of any Latin square of order... 

Let G be a graph that admits a perfect matching. A forcing set for a perfect matching M of G is a subset S of M, such that S is contained in no other perfect matching of G. This notion originally arose in chemistry in the study of molecular resonance structures. Similar concepts have been studied for block designs and graph colorings under the name defining set, and for Latin squares under the name critical set. Recently several papers have appeared on the study of forcing sets for other graph theoretic concepts such as dominating sets, orientations, and geodetics. Whilst there has been some study of forcing sets of matchings of hexagonal systems in the context of chemistry, only a few other... 

Let G be a graph that admits a perfect matching. A forcing set for a perfect matching M of G is a subset S of M, such that S is contained in no other perfect matching of G. This notion has arisen in the study of finding resonance structures of a given molecule in chemistry. Similar concepts have been studied for block designs and graph colorings under the name defining set, and for Latin squares under the name critical set. There is some study of forcing sets of hexagonal systems in the context of chemistry, but only a few other classes of graphs have been considered. For the hypercubes Qn, it turns out to be a very interesting notion which includes many challenging problems. In this paper... 

A μ-way latin trade of volume s is a set of μ partial latin rectangles (of inconsequential size) containing exactly the same s filled cells, such that if cell (i, j) is filled, it contains a different entry in each of the μ partial latin rectangles, and such that row i in each of the μ partial latin rectangles contains, set-wise, the same symbols and column j, likewise. In this paper we show that all μ-way latin trades with sufficiently large volumes exist, and state some theorems on the non-existence of μ-way latin trades of certain volumes. We also find the set of possible volumes (that is, the volume spectrum) of μ-way latin trades for μ = 4 and 5. (The case μ = 2 was dealt with by Fu,... 

The set of integers k for which there exist three latin squares of order n having precisely k cells identical, with their remaining n2 -k cells different in all three latin squares, denoted by I3[n], is determined here for all orders n. In particular, it is shown that I3[n] = {0,.,n2 -15}U [n2 - 12,n2-9,n2], for n ≫8. ©2002 Eisevier Science B.V. All rights reserved  

To study orthogonal arrays and signed orthogonal arrays, Ray-Chaudhuri and Singhi (1988 and 1994) considered some module spaces. Here, using a linear algebraic approach we define an inclusion matrix and find its rank. In the special case of Latin squares we show that there is a straightforward algorithm for generating a basis for this matrix using the so-called intercalates. We also extend this last idea. Copyright  

A μ-way Latin trade of volume s is a collection of μ partial Latin squares T1, T2,⋯, Tμ, containing exactly the same s filled cells, such that, if cell (i,j) is filled, it contains a different entry in each of the μ partial Latin squares, and such that row i in each of the μ partial Latin squares contains, set-wise, the same symbols, and column j likewise. It is called a μ-wayk-homogeneous Latin trade if, in each row and each column, Tr, for 1≤r≤μ, contains exactly k elements, and each element appears in Tr exactly k times. It is also denoted as a (μ,k,m) Latin trade, where m is the size of the partial Latin squares. We introduce some general constructions for μ-way k-homogeneous Latin... 

In a given graph G, a set S of vertices with an assignment of colors to them is a defining set of vertex coloring for G, if there exists a unique extension of the colors of 5 to a X(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,X)- Mahmoodian et al. (1999) determined the defining number of graph C3 × Cn. In this paper, we study the defining number of graph Cm × Cn, and show that Also, we prove a similar result for the defining number of graph Cn × Pn  

For a block design D, a series of block intersection graphs G i, or i-BIG(D), i = 0,..., k is defined in which the vertices are the blocks of D, with two vertices adjacent if and only if the corresponding blocks intersect in exactly i elements. A silver graph G is defined with respect to a maximum independent set of G, called an α-set. Let G be an r-regular graph and c be a proper (r + 1)-coloring of G. A vertex x in G is said to be rainbow with respect to c if every color appears in the closed neighborhood N[x] = N(x) ∪ {x}. Given an α-set I of G, a coloring c is said to be silver with respect to I if every x ∈ I is rainbow with respect to c. We say G is silver if it admits a silver... 

The chromatic number of a Latin square is the least number of partial transversals which cover its cells. This is just the chromatic number of its associated Latin square graph. Although Latin square graphs have been widely studied as strongly regular graphs, their chromatic numbers appear to be unexplored. We determine the chromatic number of a circulant Latin square, and find bounds for some other classes of Latin squares. With a computer, we find the chromatic number for all main classes of Latin squares of order at most eight  

The square of a graph G denoted by G2, is the graph with the same vertex set as G and edges linking pairs of vertices at distance at most 2 in G. The chromatic number of the square of the Cartesian product of two cycles was previously determined for some cases. In this paper, we determine the precise value of χ((Cm□Cn)2) for all the remaining cases. We show that for all ordered pairs (m,n) except for (7,11) we have χ(Cm□Cn)2)=γV((Cm□Cn)2)|α((Cm□Cn)2), where α(G) denotes the independent number of G. This settles a conjecture of Sopena and Wu (2010). We also show that the smallest integer k such that χ(Cm□Cn2)≤6 for every m,≥k is 10. This answers a question of Shao and Vesel (2013)  

An n × n matrix A is said to be silver if, for i = 1,2,...,n, each symbol in {1,2,...,2n - 1} appears either in the ith row or the ith column of A. The 38th International Mathematical Olympiad asked whether a silver matrix exists with n = 1997. More generally, a silver cube is a triple (K n d , I, c) where I is a maximum independent set in a Cartesian power of the complete graph K n , and c : V(K n d)→ {1,2,...,d(n-1)+1} is a vertex colouring where, for ν ∈ I, the closed neighbourhood N[ν] sees every colour. Silver cubes are related to codes, dominating sets, and those with n a prime power are also related to finite geometry. We present here algebraic constructions, small examples, and a... 

A t-(v,k,λ) directed design (or simply a t-(v,k, λ)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 λ elements of B. (We say that a t-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 Dt,kv, which sheds light to the existence problem for directed designs. Among 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... 

In a graph G a vertex v dominates all its neighbors and itself. A set D of vertices of G is (vertex) dominating set if each vertex of G is dominated by at least one vertex in D. The (vertex) domination number of G, denoted by γ (G), is the cardinality of a minimum dominating set of G. A set D of vertices in G is efficient dominating set if every vertex of G is dominated by exactly one vertex of D. For natural numbers n and k, where n > 2 k, a generalized Petersen graphP (n, k) is obtained by letting its vertex set be {u1, u2, ..., un} ∪ {v1, v2, ..., vn} and its edge set be the union of {ui ui + 1, ui vi, vi vi + l} over 1 ≤ i ≤ n, where subscripts are reduced modulo n. We prove a necessary... 

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. Mahmoodian and Mendelsohn (1999) 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 n0(k) and r0(k), such that for all n ≥ n0(k) and r ≥... 

A graph G is called uniquelyk-list colorable (UkLC) if there exists a list of colors on its vertices, say L= { Sv∣ v∈ V(G) } , each of size k, such that there is a unique proper list coloring of G from this list of colors. A graph G is said to have propertyM(k) if it is not uniquely k-list colorable. Mahmoodian and Mahdian (Ars Comb 51:295–305, 1999) characterized all graphs with property M(2). For k≥ 3 property M(k) has been studied only for multipartite graphs. Here we find bounds on M(k) for graphs embedded on surfaces, and obtain new results on planar graphs. We begin a general study of bounds on M(k) for regular graphs, as well as for graphs with varying list sizes. © 2018, Springer...