An orthogonal double cover of the complete graph \(K_n\) is a collection of \(n\) spanning subgraphs \(G_1, G_2, \ldots, G_n$ of \(K_n\) such that every edge of \(K_n\) belongs to exactly 2 of the \(G_i\)’s and every pair of \(G_i\)s intersect in exactly one edge.
It is proved that an orthogonal double cover exists for all \(n \geq 4\), where the \(G_i\)’s consist of short cycles; this result also proves a conjecture of Chung and West.

The induced path number of a graph \(G\) is the minimum number of subsets into which the vertex set of \(G\) can be partitioned so that each subset induces a path. The induced path number is investigated for bipartite graphs. Formulas are presented for the induced path number of complete bipartite graphs and complete binary trees. The induced path number of all wheels is determined. The induced path numbers of meshes, hypercubes, and butterflies are also considered.

Triple Youden rectangles are defined and examples are given. These combinatorial arrangements constitute a special class of \(k \times v\) row-and-column designs, \(k < v\), with superimposed treatments from three sets, namely a single set of \(v\) treatments and two sets of \(k\) treatments. The structure of each of these row-and-column designs incorporates that of a symmetrical balanced incomplete block design with \(v\) treatments in blocks of size \(k\). Indeed, when either of the two sets of \(k\) treatments is deleted from a \(k \times v\)  triple Youden rectangle, a \(k \times v\) double Youden rectangle is obtained; when both are deleted, a \(k \times v\) Youden square remains. The paper obtains an infinite class of triple Youden rectangles of size \(k \times (k+1)\). Then it presents a \(4 \times 13\) triple Youden rectangle which provides a balanced layout for two packs of playing-cards, and a \(7 \times 15\) triple Youden rectangle which incorporates a particularly remarkable \(7 \times 15\) Youden square. Triple Youden rectangles are fully balanced in a statistical as well as a combinatorial sense, and those discovered so far are statistically very efficient.

The Hall-condition number \(s(G)\) of a graph \(G\) is defined and some of its fundamental properties are derived. This parameter, introduced in [6], bears a certain relation to the chromatic number \(\chi(G)\) and the choice number \(c(G)\) (see [3] and [7]).

One result here, that \(\chi(G) – s(G)\) may be arbitrarily large, solves a problem posed in [6].

The sum of a set of graphs \(G_1,G_2,\ldots,G_k\), denoted \(\sum_{k=1}^k G_i\), is defined to be the graph with vertex set \(V(G_1)\cup V(G_2)\cup…\cup V(G_k)\) and edge set \(E(G_1)\cup E(G_2)\cup…\cup E(G_k) \cup \{uw: u \in V(G_i), w \in V(G_j) for i \neq j\}\). In this paper, the bandwidth \(B\left(\sum_{k=1}^k G_i\right)\) for \(|V(G_i)| = n_i \geq n_{i+1}=|v(G_{i+1})|,(1 \leq i < k)\) with \(B(G_1) \leq {\lceil {n_1/2}\rceil} \) is established. Also, tight bounds are given for \(B\left(\sum_{k=1}^k G_i\right)\) in other cases. As consequences, the bandwidths for the sum of a set of cycles, a set of paths, and a set of trees are obtained.

The main result of this study is that if \(p,q\) are primes such that \(q \equiv 3 (mod 4),q \leq 7,p \equiv 1 (mod 4), hef(q-1,p^{n-1} (p – 1)) =2\) and if there exists a Z-cyclic Wh(q+ 1) then a Z-cyclic Wh\(( qp^n + 1)\) exists forall \(n \geq 0\). As an ingredient sufficient for this result we prove a version of Mann’s Lemma in the ring \(Z_{qp^n}\).

In this paper we study the existence of perfect Mendelsohn designs without repeated blocks and give several general constructions. We prove that for \(k = 3\) and any \(\lambda\), and \((k,\lambda) = (4,2),(4,3)\) and \((4,4)\), the necessary conditions are also sufficient for the existence of a simple \((v,k,\lambda)\)-PMD, with the exceptions \((k,\lambda) = (6,1)\) and \((6,3)\).

A connected balanced bipartite graph \(G\) on \(2n\) vertices is almost vertex bipancyclic (i.e., \(G\) has cycles of length \(6, 8, \ldots, 2n\) through each vertex of \(G\)) if it satisfies the following property \(P(n)\): if \(x, y \in V(G)\) and \(d(x, y) = 3\) then \(d(x) + d(y) \geq n + 1\). Furthermore, all graphs except \(C_4\) on \(2n\) (\(n \geq 3\)) vertices satisfying \(P(n)\) are bipancyclic (i.e., there are cycles of length \(4, 6, \ldots, 2n\) in the graph).