A causal directed graph (CDG) is a finite directed graph with and-gates and or-nodes, in which nodes indicate true or false conditions and where arcs indicate causality. The set of all nodes implied true by a set of conditions (nodes declared true) is called the transitive closure of that set. Theorems 3-5 evaluate the number of distinct transitive closures for common CDGs. We present linear-space, linear-time algorithms for solving three transitive closure problems on CDG’s:

1. determine if a particular node is implied by a set of conditions,
2. Find the transitive closure of a set of conditions,
3. determine the minimal set of initial conditions for a given transitive closure of an acyclic CDG.

Implicit in Problem 3 is that every transitive closure of an acyclic CDG is generated by a unique minimal set of initial conditions. This is proved in Theorem 6.

A graph \(G\) is \emph{triangle-saturated} if every possible edge insertion creates at least one new triangle. Furthermore, if no proper spanning subgraph has this property, then \(G\) is minimally triangle-saturated. (Minimally triangle-saturated graphs of order \(n\) are the diameter \(2\) critical graphs when \(n \geq 3\).) The maximally triangle-free graphs of order \(n\) are a proper subset of the minimally triangle-saturated graphs of order \(n\) when \(n \geq 6\).
All triangle-saturated graphs are easily derivable from the minimally triangle-saturated graphs which are primitive, that is, have no duplicate vertices. We determine the \(23\) minimally triangle-saturated graphs of orders \(n \leq 7\) and identify the \(6\) primitive graphs among them.

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.