Let \(h \geq 1\). For each admissible \(v\), we exhibit a nested balanced path design \(H(v, 2h+1, 1)\). For each admissible odd \(v\), we exhibit a nested balanced path design \(H(v,2h,1)\). For every \(v \equiv 4 \pmod{6}\), \(v \geq 10\), we exhibit a nested balanced path design \(H(v,4,1)\) except possibly if \(v \in \{16, 52, 70\}\).

For each \(v \equiv 0 \pmod{4h}\), \(v \geq 4h\), we exhibit a nested path design \(P(v,2h+1,1)\). For each \(v \equiv 0 \pmod{4h-2}\), \(v \geq 4h-2\), we exhibit a nested path design \(P(v,2h,1)\). For every \(v \equiv 3 \pmod{6}\), \(v \geq 9\), we exhibit a nested path design \(P(v,4,1)\) except possibly if \(v = 39\).

A sequence of positive integers \(a_1 \leq a_2 \leq \ldots \leq a_n\) is called an ascending monotone wave of length \(n\), if \(a_{i+1} – a_{i} \geq a_{i} – a_{i-1}\) for \(i = 2, \ldots, n-1\). If \(a_{i+1} – a_{i} > a_{i} – a_{i-1}\) for all \(i = 2, \ldots, n-1\) the sequence is called an ascending strong monotone wave of length \(n\). Let \({Z}_k\) denote the cyclic group of order \(k\). If \(k | n\), then we define \(MW(n, {Z}_k)\) as the least integer \(m\) such that for any coloring \(f : \{1, \ldots, m\} \to {Z}_k\), there exists an ascending monotone wave of length \(n\), where \(a_n \leq m\), such that \(\sum_{i=1}^n f(a_i) = 0 \mod k\). Similarly, define \(SMW(n, {Z}_k)\), where the ascending monotone wave in \(MW(n, {Z}_k)\) is replaced by an ascending strong monotone wave. The main results of this paper are:

1. \(\frac{\sqrt{k}}{2}n \leq MW(n, Z_k) \leq c_1(k)n\). Hence, this result is tight up to a constant factor which depends only on \(k\).
2. \(\binom{n}{2} < SMW(n, {Z}_k) \leq c_2(k)n^2\). Hence, this result is tight up to a constant factor which depends only on \(k\).
3. \(MW(n, {Z}_2) = {3n}/{2}\).
4. \(\frac{23}{12}n – {7}/{6} \leq MW(n, {Z}_3) \leq 2n+3\).

These results are the zero-sum analogs of theorems proved in [1] and [5].

For \(\omega \leq 33\), the known necessary conditions for existence of a \((\nu,\{5,\omega^*\},1)\) PBD, namely \(\nu, \omega \equiv 1 \mod 4\), \(\nu \geq 4\omega+1\) and \(\nu \equiv \omega\) or \(4\omega +1 \mod 20\) are known to be sufficient in all but 26 cases. This paper provides several direct constructions which reduce the number of exceptions to 8.

The question whether every connected graph \(G\) has a spanning tree \(T\) of minimum average distance such that \(T\) is distance preserving from some vertex is answered in the negative. Moreover, it is shown that, if such a tree exists, it is not necessarily distance preserving from a median vertex.

In this note, we investigate three versions of the overfull property for graphs and their relation to the edge-coloring problem. Each of these properties implies that the graph cannot be edge-colored with \(\Delta\) colors, where \(\Delta\) is the maximum degree. The three versions are not equivalent for general graphs. However, we show that some equivalences hold for the classes of indifference graphs, split graphs, and complete multipartite graphs.

Let \(K_n\) be the complete graph on \(n\) vertices. Let \(I(X)\) denote the set of integers \(k\) for which a pair of maximum pentagon packings of graph \(X\) exist having \(k\) common 5-cycles. Let \(J(n)\) denote the set \(\{0,1,2,\ldots,P-2,P\}\), where \(P\) is the number of 5-cycles in a maximum pentagon packing of \(K_n\). This paper shows that \(I(K_n) = J(n)\), for all \(n \geq 1\).

It is shown that the Overfull Conjecture, which would provide a chromatic index characterization for a large class of graphs, and the Conformability Conjecture, which would provide a total chromatic number characterization for a large class of graphs, both in fact apply to almost all graphs, whether labelled or unlabelled. The arguments are based on Polya’s theorem, and are elementary in the sense that practically no knowledge of random graph theory is presupposed. It is similarly shown that the Biconformability Conjecture, which would provide a total chromatic number characterization for a large class of equibipartite graphs, in fact applies to almost all equibipartite graphs.

The \([0,\infty)\)-valued dominating function minimization problem has the \([0,\infty)\)-valued packing function as its linear programming dual. The standard \(\{0, 1\}\)-valued minimum dominating set problem has the \(\{0, 1\}\)-valued maximum packing set problem as its binary dual. The recently introduced complementary problem to a minimization problem is also a maximization problem, and the complementary problem to domination is the maximum enclaveless problem. This paper investigates the dual of the enclaveless problem, namely, the domination-coverage number of a graph. Specifically, let \(\eta(G)\) denote the minimum total coverage of a dominating set. The number of edges covered by a vertex \(v\) equals its degree, \(\deg v\), so \(\eta(G) = \text{MIN}\{\sum_{s \in S} \deg s: S \text{ is a dominating set}\}\). Bounds on \(\eta(G)\) and computational complexity results are presented.

In this note, we computationally prove that the size of smallest critical sets for the quaternion group of order eight, the group \(\mathbb{Z}_2 \times \mathbb{Z}_4\) and the dihedral group of order eight are 20, 21 and 22, respectively.

A graph is said \(h\)-decomposable if its edge-set is decomposable into hamiltonian cycles. In this paper, we prove that if \(G = L_1 \cup L_2 \cup L_3\) is a strongly hamiltonian bipartite cubic graph (where \(L_i\) is a perfect matching, for \(1 \leq i \leq 3\) and \((L_1, L_2, L_3)\) is a \(1\)-factorization of \(G\)), then \(G \times C_{2n+1}\) (where \(n\) is odd and \(n \geq 1\)) is decomposable. As a corollary, we show that for \(r \geq 1\) odd and \(n \geq 3\), \(K_{r,r} \times K_n\) is \(h\)-decomposable. Moreover, in the case where \(G\) is a strongly hamiltonian non-bipartite cubic graph, we prove that the same result can be derived using a special perfect matching. Hence \(K_{2r} \times K_{2n+1}\) will be \(h\)-decomposable, for \(r,n \geq 1\).

To study the product of \(G = L_1 \cup L_2 \cup L_3\) by even cycle, we define a dual graph \(G_C\) based on an alternating cycle subset of \(L_2 \cup L_3\). We show that if a non-bipartite cubic graph \(G = L_1 \cup L_2 \cup L_3\), with \(|V(G)| = 2m\), admits \(L_1 \cup L_2\) as a hamiltonian cycle and \(G_C\) is connected, then \(G \times K_2\) is hamiltonian and \(G \times C_{2n}\) has two edge-disjoint hamiltonian cycles. Finally, we prove that if \(C = L_2 \cup L_3\) and \(L_1 \cup L_3\) admits a particular alternating \(4\)-cycle \(C’\), then \(G \times C_{2n}\) is \(h\)-decomposable.