A computer program for finding knight coverings of a chessboard is described, and some improved coverings for boards of sizes \(16\times 16\) through \(25\times 25\) are shown.

Let \(G\) be a graph and \(d, d’\) be positive integers, \(d’ \geq d\). An \(m\)-\((d, d’)\)-circular distance two labeling is a function \(f\) from \(V(G)\) to \(\{0, 1, 2, \ldots, m-1\}\) such that:\(|f(u) – f(v)|_m \geq d\) if \(u\) and \(v\) are adjacent; and \(|f(u) – f(v)|_m \geq d’\) if \(u\) and \(v\) are distance two apart, where \(|x|_m := \min\{|x|, m – |x|\}\) .The minimum \(m\) such that there exists an \(m\)-\((d, d’)\)-circular labeling for \(G\) is called the \(\sigma_{d, d’}\)-number of \(G\) and denoted by \(\sigma_{d, d’}(G)\). The \(\sigma_{d, d’}\)-numbers for trees can be obtained by a first-fit algorithm. In this article, we completely determine the \(\sigma_{d, 1}\)-numbers for cycles. In addition, we show connections between generalized circular distance labeling and circular chromatic number.

The inventory of a \(2 \times m\) array \(A = A(i,j)\) consisting of \(n\) not necessarily distinct positive integers \(\mathbb{I}(2,j)\) is the \(2 \times n\) array \(\mathbb{I}(A) = \mathbb{I}(i,j)\), where \(\mathbb{I}(i,j)\) is the number of occurrences of \(\mathbb{I}(1,j)\) in \(A\). Define \(\mathbb{I}^q(A) = I(\mathbb{I}^{q-1}(A))\) for \(q \geq 1\), with \(\mathbb{I}^0(A) = A\). For every \(A\), the chain \(\{\mathbb{I}^q(A)\}\) of inventories is eventually periodic, with period \(1, 2\), or \(3\). The proof depends on runlengths of partitions of integers. A final section is devoted to an open question about cumulative inventory chains.

A decomposition \(\mathcal{F} = \{F_1, \ldots, F_r\}\) of the edge set of a graph \(G\) is called a resolving \(r\)-decomposition if for any pair of edges \(e_1\) and \(e_2\), there exists an index \(i\) such that \(d(e_1, F_i) \neq d(e_2, F_i)\), where \(d(e, F)\) denotes the distance from \(e\) to \(F\). The decomposition dimension \(dec(G)\) of a graph \(G\) is the least integer \(r\) such that there exists a resolving \(r\)-decomposition. Let \(K_n\) be the complete graph with \(n\) vertices. It is proved that \(dec(K_n) \leq \frac{1}{2} (\log_2 n)^2 (1 + o(1)).\)

For a vertex \(v\) of a graph \(G = (V, E)\), the lower independence number \(i_v(G)\) of \(G\) relative to \(v\) is the minimum cardinality of a maximal independent set in \(G\) that contains \(v\). The average lower independence number of \(G\) is \(i_{av}(G) = \frac{1}{|V|} \sum_{v\in V} i_v(G)\). In this paper, we show that if \(G\) is a tree of order \(n\), then \(i_{av}(G) \geq {2}\sqrt{n} + O(1)\), while if \(G\) is an outer-planar graph of order \(n\), then \(i_{av}(G) \geq 2\sqrt{\frac{n}{3}} + O(1)\). Both bounds are asymptotically sharp.

We consider the partition function \(b’_p(n)\), which counts the number of partitions of the integer \(n\) into distinct parts with no part divisible by the prime \(p\). We prove the following: Let \(p\) be a prime greater than \(3\) and let \(r\) be an integer between \(1\) and \(p-1\), inclusively, such that \(24r+1\) is a quadratic nonresidue modulo \(p\). Then, for all nonnegative integers \(n\), \(b’_p{(pn+r)} \equiv 0 \pmod{2}.\)

We show that:(a) the special product of two cycles is Hamiltonian decomposable, and (b) if \(G_1\) and \(G_2\) are two Hamiltonian decomposable graphs and at least one of their complements is Hamiltonian decomposable, then the special product of \(G_1\) and \(G_2\) is Hamiltonian decomposable.

A vertex-magic total labeling on a graph \(G\) is a one-to-one map \(\lambda\) from \(V(G) \cup E(G)\) onto the integers \(1, 2, \ldots, |V(G) \cup E(G)|\) with the property that, given any vertex \(x\), \(\lambda(x) + \sum_{y \sim x} \lambda(y) = k\) for some constant \(k\).

In this paper, we completely determine which complete bipartite graphs have vertex-magic total labelings.

In this paper, the notions of \(c\)-Motzkin and \(d\)-Motzkin words are introduced, studied, and the cardinal numbers of their sets are evaluated. Finally, bijections between the sets of the introduced Motzkin words and certain sets of noncrossing partitions are exhibited.

Vizing conjectured that \(\gamma(G)\gamma(H) \leq \gamma(G \Box H)\) for all graphs \(G\) and \(H\), where \(\gamma(G)\) denotes the domination number of \(G\) and \(G \Box H\) is the Cartesian product of \(G\) and \(H\). We prove that if \(G\) and \(H\) are \(\delta\)-regular, then, with only a few possible exceptions, Vizing’s conjecture holds. We also prove that if \(\delta(G), \Delta(G), \delta(H)\), and \(\Delta(H)\) are in a certain range, then Vizing’s conjecture holds. In particular, we show that for graphs of order at most \(n\) with minimum degrees at least \(\sqrt{n} \ln n\), the conjecture holds.