Possible volumes of t-(v, t + 1) Latin trades

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 \Ts{t}{v}{k}. 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 $s_i=2^{t+1}-2^{(t+1)-i}$, $0\leq i\leq t+1$, as critical points and then we show that $s_i\in S(t,k)$, for any $0\leq i\leq t+1$, and if $s\in (s_i,s_{i+1}),0\leq i\leq t$, then $s\notin S(t,t+1)$. As an example, we determine S(3,4) precisely.Comment: 12 page

A new bound on the size of the largest critical set in a Latin square

A critical set in an n x n array is a set C of given entries, such that there exists a unique extension of C to an n x n Latin square and no proper subset of C has this property. The cardinality of the largest critical set in any Latin square of order n is denoted by lcs(n). In 1978 Curran and van Rees proved that lcs(n) <= n^2 - n. Here we show that lcs(n) <= n^2-3n+3.Comment: 10 pages, LaTe

Totally Silver Graphs

A totally silver coloring of a graph G is a k--coloring of G such that for every vertex v \in V(G), each color appears exactly once on N[v], the closed neighborhood of v. A totally silver graph is a graph which admits a totally silver coloring. Totally silver coloring are directly related to other areas of graph theory such as distance coloring and domination. In this work, we present several constructive characterizations of totally silver graphs and bipartite totally silver graphs. We give several infinite families of totally silver graphs. We also give cubic totally silver graphs of girth up to 10

On uniquely list colorable graphs

Let G be a graph with n vertices and suppose that for each vertex v in G, there exists a list of k colors L(v), such that there is a unique proper coloring for G from this collection of lists, then G is called a uniquely k-list colorable graph. Recently M. Mahdian and E.S. Mahmoodian characterized uniquely 2-list colorable graphs. Here we state some results which will pave the way in characterization of uniquely k-list colorable graphs. There is a relationship between this concept and defining sets in graph colorings and critical sets in latin squares.Comment: 13 page

On the existence of k-homogeneous Latin bitrades

Following the earlier work on {homogeneous Latin bitrades by Cavenagh, Donovan, and Dr'apal (2003 and 2004) Bean, Bidkhori, Khosravi, and E. S. Mahmoodian (2005) we prove the following results. All k-homogeneous Latin bitrades of volume km exist, for 1) all odd number k and m is greater than or equal to k, 2) all even number k > 2 and m is greater than or equal to min{k +u,3k/2}, where u is any odd number which divides k, 3) all m is greater than or equal to k, where k is greater than or equal to 3 and is lesser than or equal to 37

The Chromatic Number of Finite Group Cayley Tables

The chromatic number of a latin square $L$, denoted $\chi(L)$, is the minimum number of partial transversals needed to cover all of its cells. It has been conjectured that every latin square satisfies $\chi(L) \leq |L|+2$. If true, this would resolve a longstanding conjecture---commonly attributed to Brualdi---that every latin square has a partial transversal of size $|L|-1$. Restricting our attention to Cayley tables of finite groups, we prove two main results. First, we resolve the chromatic number question for Cayley tables of finite Abelian groups: the Cayley table of an Abelian group $G$ has chromatic number $|G|$ or $|G|+2$, with the latter case occurring if and only if $G$ has nontrivial cyclic Sylow 2-subgroups. Second, we give an upper bound for the chromatic number of Cayley tables of arbitrary finite groups. For $|G|\geq 3$, this improves the best-known general upper bound from $2|G|$ to $\frac{3}{2}|G|$, while yielding an even stronger result in infinitely many cases

On the star arboricity of hypercubes

A Hypercube $Q_n$ is a graph in which the vertices are all binary vectors of length n, and two vertices are adjacent if and only if their components differ in exactly one place. A galaxy or a star forest is a union of vertex disjoint stars. The star arboricity of a graph $G$, ${\rm sa}(G)$, is the minimum number of galaxies which partition the edge set of $G$. In this paper among other results, we determine the exact values of ${\rm sa}(Q_n)$ for $n \in \{2^k-3, 2^k+1, 2^k+2, 2^i+2^j-4\}$, $i \geq j \geq 2$. We also improve the last known upper bound of ${\rm sa}(Q_n)$ and show the relation between ${\rm sa}(G)$ and square coloring.Comment: Australas. J. Combin., vol. 59 pt. 2, (2014

A linear algebraic approach to orthogonal arrays and Latin squares

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.Comment: 11 page

Silver block intersection graphs of Steiner 2-designs

For a block design $\cal{D}$, a series of {\sf block intersection graphs} $G_i$, or $i$-{\rm BIG}($\cal{D}$), $i=0, ..., k$ is defined in which the vertices are the blocks of $\cal{D}$, 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 a {\sf diagonal} of that graph. Let $G$ be $r$-regular and $c$ be a proper $(r + 1)$-coloring of $G$. A vertex $x$ in $G$ is said to be {\sf rainbow} with respect to $c$ if every color appears in the closed neighborhood $N[x] = N(x) \cup \{x\}$. Given a diagonal $I$ of $G$, a coloring $c$ is said to be silver with respect to $I$ if every $x\in I$ is rainbow with respect to $c$. We say $G$ is {\sf silver} if it admits a silver coloring with respect to some $I$. We investigate conditions for 0-{\rm BIG}($\cal{D}$) and 1-{\rm BIG}($\cal{D}$) of Steiner systems ${\cal{D}}=S(2,k,v)$ to be silver

On the chromatic number of Latin square graphs

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