In a recent paper, Gustavus J. Simmons introduced a new class of combinatorial-geometric objects he called “campaign graphs”. A \(k\)-campaign graph is a collection of points and segments such that each segment contains precisely \(k\) of the points, and each point is the endpoint of precisely one segment. Among other results, Simmons proved the existence of infinitely many critical \(k\)-campaign graphs for \(k \leq 4\).

The main aim of this note is to show that Simmons’ result holds for \(k = 5\) and \(6\) as well, thereby providing proofs, amplifications and a correction for statements of this author which Dr. Simmons was kind enough to include in a postscript to his paper.

Let \(P(G)\) denote the chromatic polynomial of a graph \(G\). Two graphs \(G\) and \(H\) are chromatically equivalent, writen \(G \sim H\), if \(P(G) = P(H)\). A graph \(G\) is chromatically unique if \(G \cong H\) for any graph H such that \(H \sim G\). Let \(\mathcal{G}\) denote the class of \(2\)-connected graphs of order n and size \(n+ 2\) which contain a \(4\)-cycle or two triangles. It follows that if \(G \in \mathcal{G}\) and \(H \sim G\),then \(H \in \mathcal{G}\). In this paper, we determine all equivalence classes in \(\mathcal{G}\) under the equivalence relation \(‘\sim’\) and characterize the structures of the graphs in each class. As a by-product of these,we obtain three new families of chromatically unique graphs.

We show that for all odd \(m\), there exists a directed \(m\)-cycle system of \(D_n\) that has an \(\left\lfloor \frac{m}{2} \right\rfloor\)-nesting, except possibly when \(n \in \{3m+1, 6m+1\}\).

Given an overlarge set of Steiner triple systems, each on \(v\) points, we construct an overlarge set of Steiner triple systems, each on \(2v+1\) points. Overlarge sets with specified properties can be constructed in this way; in particular, we construct overlarge sets which cannot be derived from Steiner quadruple systems.

Halberstam, Hoffman and Richter introduced the idea of a Latin triangle as an analogue of a Latin square, showed the existence or non-existence of Latin triangles for small orders, and used a multiplication technique to generate triangles of orders \(3^n\) and \(3^n – 1\). We generalize this multiplication theorem and provide a construction of Latin triangles of odd order \(n\) for \(n\) such that \(n+2\) is prime. We also discuss scalar multiplication, orthogonal triangles, and results of computer searches.

A graph covering projection is a local graph homeomorphism. Certain partitions of the vertex set of the preimage graph induce a notion of “concreteness”. The concrete graph covering projections will be counted up to isomorphism.

The set of all distinct blocks of a \(t\)-design is referred to as the support of the design and its cardinality is denoted by \(b^*\). In this article (i) the set of all possible \(b^*\)’s for the case of \(3\)-\((8,4,\lambda)\) designs is determined and for each feasible \(b^*\) a design with a minimum \(b\) is produced;(ii) it is shown that a \(2\)-\((8,4,3\lambda)\) design is a \(3\)-\((8,4,\lambda)\) design if and only if it is self-complementary; (iii) it is shown that there are at least \(63\) pairwise non-isomorphic \(3\)-\((8,4,5)\) designs.

The cycle graph \(C(G)\) of a graph \(G\) has vertices which correspond to the chordless cycles of \(G\), and two vertices of \(C(G)\) are adjacent if the corresponding chordless cycles of \(G\) have at least one edge in common. If \(G\) has no cycle, then we define \(C(G)=\emptyset\), the empty graph. For an integer \(n \geq 2\), we define recursively the \(n\)-th iterated cycle graph \(C^n(G)\) by \(C^n(G)=C(C^{n-1}(G))\). We classify graphs according to their cycle graphs as follows. A graph \(G\) is \emph{cycle-vanishing} if there exists an integer \(n\) such that \(C^n(G)=\emptyset\); and \(G\) is \emph{cycle-periodic} if there exist two integers \(n\) and \(p \geq 1\) such that \(C^{n+p}(G)\cong C^n(G) \neq \emptyset\). Otherwise, \(G\) is cycle-expanding. We characterize these three types of graphs, and give some other results on cycle graphs.

A Latin square of order \(n\) is an \(n \times n\) array such that each of the integers \(1, 2, \ldots, n\) (or any set of \(n\) distinct symbols) occurs exactly once in each row and each column. A Latin square \(L = [l_{i,j}]\) is said to be \underline{commutative} provided that \(l_{i,j} = l_{j,i}\) for all \(i\) and \(j\). Two Latin squares, \(L = [l_{i,j}]\) and \(M = [m_{i,j}]\), are said to have \underline{intersection} \(k\) if there are exactly \(k\) cells \((i,j)\) such that \(l_{i,j} = m_{i,j}\).

Let \(I[n] = \{0, 1, 2, \ldots, n^2-9, n^2-8, n^2-7, n^2-6,n^2\}\), \(H[n] = I[n] \cup \{n^2-7, n^2-4\}\), and \(J[n]\) be the set of all integers \(k\) such that there exists a pair of commutative Latin squares of order \(n$ which have intersection \(k\). In this paper, we prove that \(J[n] = I[n]\) for each odd \(n \geq 7\), \(J[n] = H[n]\) for each even \(n \geq 6\), and give a list of \(J[n]\) for \(n \leq 5\). This totally solves the intersection problem of two commutative Latin squares.