We show that, for all primes \(p \equiv 1 \pmod{4}\), \(29 \leq p < 10,000\), \(p \neq 97, 193, 257, 449, 641, 769, 1153, 1409, 7681\), there exist \({Z}\)-cyclic triplewhist tournaments on \(p\) elements which are also Mendelsohn designs. We also show that such designs exist on \(v\) elements whenever \(v\) is a product of such primes \(p\).

An algorithm is presented in which a polynomial deck, \(\mathcal{P}D\), consisting of \(m\) polynomials of degree \(m-1\), is analysed to check whether it is the deck of characteristic polynomials of the one-vertex-deleted subgraphs of the line graph, \(H\), of a triangle-free graph, \(G\). We show that if two necessary conditions on \(\mathcal{P}D\), identified by counting the edges and triangles in \(H\), are satisfied, then one can construct potential triangle-free root graphs, \(G\), and by comparing the polynomial decks of the line graph of each with \(\mathcal{P}D\), identify the root graph.

Let \(\sigma_2(G) = \min\{d_G(u)+d_G(v) | u,v \in V(G), u,v \notin E(G)\}\) for a non-complete graph \(G\). An \([a, b]\)-factor of \(G\) is a spanning subgraph \(F\) with minimum degree \(\delta(F) \geq a\) and maximum degree \(\Delta(F) \leq b\).
In this note, we give a partially positive answer to a conjecture of M. Kano. We prove the following results:

Let \(G\) be a 2-edge-connected graph of order \(n\) and let \(k \geq 2\) be an integer. If \(\sigma_2(G) \geq {4n}/{(k +2)}\), then \(G\) has a 2-edge-connected \([2, k]\)-factor if \(k\) is even and a 2-edge-connected \([2, k + 1]\)-factor if \(k\) is odd.
Indeed, if \(k\) is odd, there exists a graph \(G\) which satisfies the same hypotheses and has no 2-edge-connected \([2, k]\)-factor.
Nevertheless, we have shown that if \(G\) is 2-connected with minimum degree \(\delta(G) \geq {2n}/{(k +2)}\), then \(G\) has a 2-edge-connected \([2, k]\)-factor.

The Ramsey numbers \(r(C_4,G)\) are determined for all graphs \(G\) of order six.

In a \(t-(\nu,k,\lambda)\) directed design, the blocks are ordered \(k\)-tuples and every ordered \(t\)-tuple of distinct points occurs in exactly \(\lambda\) blocks (as a subsequence). We show that a simple \(3-(\nu,4,2)\) directed design exists for all \(v\). This completes the proof that the necessary condition \(\lambda v\equiv 0 \pmod 2\) for the existence of a \(3-(\nu,4,\lambda)\) directed design is sufficient.

We give a conjecture for the total chromatic number \(\chi_T\) of all Steiner systems and show its relationship to the celebrated Erdős, Faber, Lovász conjecture. We show that our conjecture holds for projective planes, resolvable Steiner systems and cyclic Steiner systems by determining their total chromatic number.

We propose a number of problems about \(r\)-factorizations of complete graphs. By a completely novel method, we show that \(K_{2n+1}\) has a \(2\)-factorization in which all \(2\)-factors are non-isomorphic. We also consider \(r\)-factorizations of \(K_{rn+1}\) where \(r \geq 3\); we show that \(K_{rn+1}\) has an \(r\)-factorization in which the \(r\)-factors are all \(r\)-connected and the number of isomorphism classes in which the \(r\)-factors lie is either \(2\) or \(3\).