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).

Let \(T(m,n)\) denote the number of \(m \times n\) rectangular standard Young tableaux with the property that the difference of any two rows has all entries equal. Let \(T(n) = \sum\limits_{d|n} T(d,n/d)\). We find recurrence relations satisfied by the numbers \(T(m,n)\) and \(\hat{T}(n)\), compute their generating functions, and express them explicitly in some special cases.

A labeling (function) of a graph \(G\) is an assignment \(f\) of nonnegative integers to the vertices of \(G\). Such a labeling of \(G\) induces a labeling of \(L(G)\), the line graph of \(G\), by assigning to each edge \(uv\) of \(G\) the label \(\lvert f(u) – f(v)\rvert\). In this paper we investigate the iteration of such graph labelings.