In this paper, we derive some inequalities on the existence of balanced arrays (B-arrays) of strength six and with two symbols by using some results involving moments from Statistics. Besides providing illustrative examples, we will make brief comments on the use of these combinatorial arrays in Statistical Design of Experiments.

We estimate the number of labelled loop-free Eulerian oriented graphs with multiple edges with \(n\) vertices by using an \(n\)-dimensional Cauchy integral. An asymptotic formula is obtained for the case where the multiplicity of each edge is \(O(\log n)\).

The number of hypohamiltonian and that of hypotraceable \(n\)-vertex digraphs are both bounded below by a superexponential function of \(n\) for \(n \geq 6\) and \(n \geq 7\), respectively.

In a graph \(G = (V, E)\), a set \(S \subset V\) is a dominating set if each vertex of \(V – S\) is adjacent to at least one vertex in \(S\).

Approximately 1000 papers have been written on domination-related concepts, with more than half of them appearing in the literature in the last five years. Obviously, a comprehensive survey is beyond the scope of this paper, so a brief overview is presented.

Let \(G\) be a cubic graph containing no subdivision of the Petersen graph. If \(G\) has a \(2\)-factor \(F\) consisting of two circuits \(C_1\) and \(C_2\) such that \(C_1\) is chordless and \(C_2\) has at most one chord, then \(G\) is edge-\(3\)-colorable.

This result generalizes an early result by Ellingham and is a partial result of Tutte’s edge-\(3\)-coloring conjecture.

Let \(f(n,k)\) be the maximum chromatic number among all graphs whose edge set can be covered by \(n\) copies of \(K(n)\), the complete graph on \(n\) vertices, so that any two of those \(K(n)\) share at most \(k\) vertices.

It has been known that \(f(n,k) = (1 – o(1)).n^{{3}/{2}}\) for \(k \geq n^{{1}/{2}}\). We show that
\[(1 – o(1))n.k \leq f(n,k) \leq (k+1)(n-k)\]
for \(k < n^{{1}/{2}}\), hence, for \({1}/{k} = o(1)\), \[f(n,k) = (1 + o(1))n.k.\]

A string is strongly square-free if it contains no Abelian squares; that is, adjacent substrings which are permutations of each other. We discuss recent results concerning the construction of strongly square-free finite strings.

It is shown that the circuit polynomial characterizes many of the well-known families of graphs. These include chains, stars, cycles, complete graphs, regular complete bipartite graphs, and wheels. Some analogous results are deduced for the characteristic polynomial and the \(\mu\)-polynomial.

A connected dominating set is a dominating set \(S\) with the additional property that the subgraph induced by \(S\) is connected. We are interested in the collection C of graphs in which every minimal connected dominating set is of one size.
Trees, for instance, clearly belong to this collection. A partial characterization will be discussed; in particular, we determine those graphs which have the property that all spanning trees have the same number of leaves. It is noted that membership in this sub-collection of C can be determined in polynomial time.

A continuum with finitely many non-cut points is an irreducible tree. A two-variable power series for the number of (unlabelled) irreducible trees with \(p\) pendant and \(q\) interior vertices.


The result is then specialized to get Harary's series for the number of irreducible trees with \(n\) vertices and to another series for the number of irreducible trees with \(p\) pendant vertices, a result of interest in continuum theory.