In this paper we give a partial answer to a query of Lindner conceming the quasigroups arising from \(2\)-perfect \(6\)-cycle systems.

Consider the paths \(\pi_t(i_1), \ldots, \pi_t(i_k)\) from the root to the leaves \(i_1, \ldots, i_k\) in a random binary tree \(t\) with \(n\) internal nodes, where all such trees are assumed equally likely and the leaves are enumerated from left to right. We investigate, for fixed \(i_1, \ldots, i_k\) and \(n\), the average size of \(\pi_t(i_1)\cup \ldots \cup \pi_t(i_k)\) resp. of \(\pi_t(i_1)\cap \ldots \cap \pi_t(i_k)\) (the latter corresponding to the average depth of the smallest subtree containing \(i_1, \ldots, i_k\)). By a rotation argument, both problems are reduced to the case \(k=1\), for which a solution is known. Furthermore, formulas for the probability distributions of the depth of leaf \(i\), the distance between leaf \(i\) and \(j\) and the length of \(\pi_t(i) \cap \pi_t(j)\) are derived.

Chetwynd and Hilton made the following \({edge-colouring \; conjecture}\): if a simple graph \(G\) satisfies \(\Delta(G) > \frac{1}{3}|V(G)|\), then \(G\) is Class \(2\) if and only if it contains an overfull subgraph \(H\) with \(\Delta(H) = \Delta(G)\). They also made the following \({total-colouring \; conjecture}\): if a simple graph \(G\) satisfies \(\Delta(G) \geq \frac{1}{2}(|V(G)|+ 1)\), then \(G\) is Type \(2\) if and only if \(G\) contains a non-conformable subgraph \(H\) with \(\Delta(H) = \Delta(G)\). Here we show that if the edge-colouring conjecture is true for graphs of even order satisfying \(\Delta(G) > \frac{1}{2}|V(G)|\), then the total-colouring conjecture is true for graphs of odd order satisfying \(\delta(G) \geq \frac{3}{4}{|V(G)|} – \frac{1}{4}\) and \(\text{def}(G) \geq 2(\Delta(G) – \delta(G) + 1)\).

We correct an omission by Mathon in his classification of symmetric \((31, 10, 3)\)-designs with a non-trivial automorphism group and find that there are a further six such designs, all with an automorphism group of order \(3\).

A \({dominating \; function}\) is a feasible solution to the LP relaxation of the minimum dominating set \(0-1\) integer program. A minimal dominating function (MDF) g is called universal if every convex combination of g and any other MDF is also a MDF. The problem of finding a universal MDF in a tree \({T}\) can also be described by a linear program. This paper describes a linear time algorithm that finds a universal MDF in \({T}\), if one exists.

Let \(H\) be a digraph whose vertices are called colours. Informally, an \(H\)-colouring of a digraph \(G\) is an assignment of these colours to the vertices of \(G\) so that adjacent vertices receive adjacent colours. In this paper we continue the study of the \(H\)-colouring problem, that is, the decision problem “Does there exist an \(H\)-colouring of a given digraph \(G\)?”. In particular, we prove that the \(H\)-colouring problem is NP-complete if the digraph \(H\) consists of a directed cycle with two chords, or two directed cycles joined by an oriented path, or is obtained from a directed cycle by replacing some arcs by directed two-cycles, so long as \(H\) does not retract to a directed cycle. We also describe a new reduction which yields infinitely many new infinite families of NP-complete \(H\)-colouring problems.

Bondy conjectures that if \(G\) is a \(2\)-edge-connected simple graph with \(n\) vertices, then at most \((2n-1)/{3}\) cycles in \(G\) will cover \(G\). In this note, we show that if \(G\) is a plane triangulation with \(n \geq 6\) vertices, then at most \((2n-3)/{3}\) cycles in \(G\) will cover \(G\).

An affine (respectively projective) failed design \(D\), denoted by \(\text{AFD}(q)\) (respectively \(\text{PFD}(q)\)) is a configuration of \(v = q^2\) points, \(b = q^2 + q + 1\) blocks and block size \(k = q\) (respectively \(v = q^2 + q + 1\) points, \(b = q^2 + q + 2\) blocks and block size \(k = q + 1\)) such that every pair of points occurs in at least one block of \(D\) and \(D\) is minimal, that is, \(D\) has no block whose deletion gives an affine plane (respectively a projective plane) of order \(q\). These configurations were studied by Mendelsohn and Assaf and they conjectured that an \(\text{AFD}(q)\) exists if an affine plane of order \(q\) exists and a \(\text{PFD}(q)\) never exists. In this paper, it is shown that an \(\text{AFD}(5)\) does not exist and, therefore, the first conjecture is false in general, \(\text{AFD}(q^2)\) exists if \(q\) is a prime power and the second conjecture is true, that is, \(\text{PFD}(q)\) never exists.