Sixteen non-isomorphic symmetric \(2\)-\((31, 10, 3)\) designs with trivial full automorphism group are constructed.

We define a sequence of positive integers \({A} = (a_1, \ldots, a_n)\) to be a count-wheel of length \(n\) and weight \(w = a_1 + \cdots + a_n\) if it has the following property:
Let \(\overline{A}\) be the infinite sequence \((\overline{a_i})=(a_1, \ldots, a_n, a_1, \ldots, a_n, \ldots)\). Then there is a sequence \(0 = i(0) < i(1) < i(2) < \cdots\) such that for every positive integer \(k\), \(\overline{a}_{i(k-1)+1} + \cdots + \overline{a}_{i(k)} = k\). There are obvious notions of when a count-wheel is reduced or primitive. We show that for every positive integer \(w\), there is a unique reduced count-wheel of weight \(w\), denoted \([w]\). Also, \([w]\) is primitive if and only if \(w\) is odd. Further, we give several algorithms for constructing \([w]\), and a formula for its length. (Remark: The count-wheel \([15] = (1, 2, 3, 4, 3, 2)\) was discovered by medieval clock-makers.)

We present 3 connections between the two nonisomorphic \(C(6, 6, 1)\) designs and the exterior lines of an oval in the projective plane of order four. This connection demonstrates the existence of precisely four nonisomorphic large sets of \(C(6, 6, 1)\) designs.

Using computer algorithms we found that there exists a unique, up to isomorphism, graph on \(21\) points and \(125\) graphs on \(20\) points for the Ramsey number \(R(K_5 – e, K_5 – e) = 22\). We also construct all graphs on \(n\) points for the Ramsey number \(R(K_4 – e, K_5 – e) = 13\) for all \(n \leq 12\).

Affine \((\mu_1,\ldots,\mu_t)\)-resolvable \((\tau,\lambda)\)-designs are introduced. Constructions of such designs are presented.

Using basis reduction, we settle the existence problem for \(4\)-\((21,5,\lambda)\) designs with \(\lambda \in \{3,5,6,8\}\). These designs each have as an automorphism group the Frobenius group \(G\) of order \(171\) fixing two points. We also show that a \(4\)-\((21,5,1)\) design cannot have the subgroup of order \(57\) of \(G\) as an automorphism group.

A finite group is called \(P_n\)-sequenceable if its nonidentity elements can be listed \(x_1, x_2, \ldots, x_{k}\) such that the product \(x_i x_{i+1} \cdots x_{i+n-1}\) can be rewritten in at least one nontrivial way for all \(i\). It is shown that \(S_n, A_n, D_n\) are \(P_3\)-sequenceable, that every finite simple group is \(P_4\)-sequenceable, and that every finite group is \(P_5\)-sequenceable. It is conjectured that every finite group is \(P_3\)-sequenceable.

In this paper, we give two constructive proofs that all \(4\)-stars are Skolem-graceful. A \(4\)-star is a graph with 4 components, with at most one vertex of degree exceeding 1 per component. A graph \(G = (V, E)\) is Skolem-graceful if its vertices can be labelled \(1, 2, \ldots, |V|\) so that the edges are labelled \(1, 2, \ldots, |E|\), where each edge-label is the absolute difference of the labels of the two end-vertices. Skolem-gracefulness is related to the classic concept of gracefulness, and the methods we develop here may be useful there.

We consider two seemingly related problems. The first concerns pairs of graphs \(G\) and \(H\) containing endvertices (vertices of degree \(1\)) and having the property that, although they are not isomorphic, they have the same collection of endvertex-deleted subgraphs.

The second question concerns graphs \(G\) containing endvertices and having the property that, although no two endvertices are similar, any two endvertex-deleted subgraphs of \(G\) are isomorphic.