This note gives what is believed to be the first published example of a symmetric \(11 \times 11\) Latin square which, although not cyclic, has the property that the permutation between any two rows is an \(11\)-cycle. The square has the further property that two subsets of its rows constitute \(5 \times 11\) Youden squares. The note shows how this \(11 \times 11\) Latin square can be obtained by a general construction for \(n \times n\) Latin squares where \(n\) is prime with \(n \geq 11\). The permutation between any two rows of any Latin square obtained by the general construction is an \(n\)-cycle; two subsets of \((n-1)/2\) rows from the Latin square constitute Youden squares if \(n \equiv 3 \pmod{8}\).

The twenty-five year old \(\lambda\)-design conjecture remains unsettled. Attempts to characterize these irregular, tight, \(2\)-designs have produced a great number of parametric and dual structure characterizations of the so-called Type-I Designs. We establish some new structural characterizations and establish the conjecture in the smallest unsettled case (\(\lambda = 14\)) of the \(2p\) family.

In this paper we consider a random walk in a plane in which a particle at any stage moves one unit in any one of the four directions, namely, north, south, east, and west with equal probability and derive the joint and marginal distributions of certain characteristics of this random walk by using combinatorial methods.

A subset \(S\) of an ordered set \(P\) is called a cutset if each maximal chain of \(P\) has nonempty intersection with \(S\); if, in addition, \(S\) is also an antichain, it is an antichain cutset. We consider new characterizations and generalizations of these and related concepts. The main generalization is to make our definitions in graph theoretic terms. For instance, a cutset is a subset \(S\) of the vertex set \(V\) of graph \(G = (V, E)\) which meets each extremal path of \(G\). Our principal results include (1)a characterization, by means of a closure property, of those antichains which are cutsets;(2) a characterization, by means of “forbidden paths” in the graph, of those graphs which can be expressed as the union of antichain cutsets;(3) a simpler proof of an existing result about \(N\)-free orders; and (4) efficient algorithms for many related problems, such as constructing antichain cutsets containing or excluding specified elements or forming a chain.

We include a brief discussion of the use of antichain cutsets in a parsing problem for \(LR(k)\) languages.

The \(n\)-star graph \(S_n\) is a simple graph whose vertex set is the set of all \(n!\) permutations of \(\{1,2,\ldots,n\}\) and two vertices \(\alpha\) and \(\beta\) are adjacent if and only if \(\alpha(1) \neq \beta(1)\) and \(\alpha(i) \neq \beta(i)\) for exactly one \(i\), \(i \neq 1\). In the paper, we determine the values of the domination number \(\gamma\), the independent domination number \(\gamma_i\), the perfect domination number \(\gamma_p\), and we obtain bounds for the total domination number \(\gamma_t\) and the connected domination number \(\gamma_c\) for \(S_n\).

Holey factorizations of \(K_{v_1,v_2,\ldots,v_n}\) are a basic building block in the construction of Room frames. In this paper we give some necessary conditions for the existence of holey factorizations and give a complete enumeration for nonisomorphic sets of orthogonal holey factorizations of several special types.

It is shown that the maximal number of pairwise edge disjoint forests, \(F\), of order six in the complete graph \(K_n\), and the minimum number of forests of order six, whose union is \(K_n\) are \(\lfloor\frac{n(n-1)}{2e(F)}\rfloor\) and \(\lceil\frac{n(n-1)}{2e(F)}\rceil\), \(n\geq 6\), respectively and \(e(F)\) is the number of edges of \(F\). (\(\lfloor x\rfloor\) denotes the largest integer not exceeding \(x\) and \(\lceil x\rceil\) the least integer not less than \(x\)). Some generalizations to multiple copies of these forests and of paths are also given.

We study four \(q\)-series. Each of which is interpreted combinatorially in three different ways. This results in four new classes of infinite \(3\)-way partition identities. In some particular cases we get even \(4\)-way partition identities. Our every \(3\)-way identity gives us three Roderick-Ramanujan type identities and \(4\)-way identity gives six. Several partition identities due to Gordon \((1965)\), Hirschhorn \((1979)\), Subbarao \((1985)\), Blecksmith et al. \((1985)\), Agarwal \((1988)\) and Subbarao and Agarwal (1988) are obtained as particular cases of our general theorems.

Motivated by the spectral radius of a graph, we introduce the notion of numerical radius for multigraphs and directed multigraphs, and it is proved that, unlike the spectral radius, the numerical radius is invariant under changes in the orientation of a directed multigraph. An analogue of the Perron-Frobenius theorem is given for the numerical radius of a matrix with nonnegative entries.

We consider the polytope \(\mathcal{P}(s)\) of generalized tournament matrices with score vector \(s\). For the case that \(s\) has integer entries, we find the extreme points of \(\mathcal{P}(s)\) and discuss the graph-theoretic structure of its \(1\)-skeleton.