It is shown that if a graph \(G\) is connected, claw-free, and such that the vertices of degree 1 of every induced bull have a common neighbor in \(G\), then \(G\) is traceable.

Some extremal set problems can be phrased as follows. Given an \(m \times n\) \((0,1)\)-matrix \(A\) with no repeated columns and with no submatrix of a certain type, what is a bound on \(n\) in terms of \(m\)? We examine a conjecture of Frankl, Füredi, and Pach and the author that when we forbid a \(k \times l\) submatrix \(F\) then \(n\) is \(O(m^{k})\). Two proof techniques are presented, one is amortized complexity and the other uses a result of Alon to show that \(n\) is \(O(m^{2k-1-\epsilon})\) for \(\epsilon=(k-1)/(13 \log_2 l)\), improving on the previous bound of \(O(m^{2k-1})\).

A graph \(H\) is \(G\)-decomposable if \(H\) can be decomposed into subgraphs, each of which is isomorphic to \(G\). A graph \(G\) is a greatest common divisor of two graphs \(G_1\) and \(G_2\) if \(G \) is a graph of maximum size such that both \(G_1\) and \(G_2\) are \(G\)-decomposable. The greatest common divisor index of a graph \(G\) of size \(q\) is the greatest positive integer \(n\) for which there exist graphs \(G_1\) and \(G_2\), both of size at least \(nq\), such that \(G\) is the unique greatest common divisor of \(G_1\) and \(G_2\). The corresponding concepts are defined for digraphs. Relationships between greatest common divisor index for a digraph and for its underlying graph are studied. Several digraphs are shown to have infinite index, including matchings, short paths, union of stars, transitive tournaments, the oriented 4-cycle. It is shown that for \(5 \leq p \leq 10\), if a graph \(F\) of sufficiently large size is \(C_p\)-decomposable, then \(F\) is also \((P_{p-1} \cup P_3)\)-decomposable. From this it follows that the even cycles \(C_6\), \(C_8\) and \(C_{10}\) have finite greatest common divisor index.

A chess-like game board called a hive, consisting of hexagonal cells, and a board piece called a queen are defined. For queens on hexagonally shaped hives, values are obtained for the lower and independent domination numbers, the upper independence number and the diagonal domination number, as well as a lower bound for the upper domination number. The concept of a double column placement is introduced.

Two vertices in a graph \(H\) are said to be pseudosimilar if \(H – u\) and \(H – v\) are isomorphic but no automorphism of \(H\) maps \(u\) into \(v\). Pseudosimilar edges are analogously defined. Graphs in which every vertex is pseudosimilar to some other vertex have been known to exist since 1981. Producing graphs in which every edge is pseudosimilar to some other edge proved to be more difficult. We here look at two constructions of such graphs, one from \(\frac{1}{2}\)-transitive graphs and another from edge-transitive but not vertex-transitive graphs. Some related questions on Cayley line-graphs are also discussed.

The maximum cardinality of a partition of the vertex set of a graph \(G\) into dominating sets is the domatic number of \(G\), denoted \(d(G)\). The codomatic number of \(G\) is the domatic number of its complement, written \({d}(\overline{G})\). We show that the codomatic number for any cubic graph \(G\) of order \(n\) is \(n/2\), unless \(G \in \{K_4, G_1\}\) where \(G_1\) is obtained from \(K_{2,3} \cup K_3\) by adding the edges of a 1-factor between \(K_3\) and the larger partite set of \(K_{2,3}\).

Various connections have been established between the permanent and the determinant of the adjacency matrix of a graph. Connections are also made between these scalars and the number of perfect matchings in a graph. We establish conditions for graphs to have determinant 0 or \(\pm1\). Necessary conditions and sufficient conditions are obtained for graphs to have permanent equal to 0 or to 1.

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.