A graceful labeling of a graph \( G \) of size \( n \) is an assignment of labels from \( \{0, 1, \ldots, n\} \) to the vertices of \( G \) such that when each edge has assigned a weight defined by the absolute difference of its end-vertices, the resulting weights are distinct. The gracefulness of a graph \( G \) is the smallest positive integer \( k \) for which it is possible to label the vertices of \( G \) with distinct elements from the set \( \{0, 1, \ldots, k\} \) in such a way that distinct edges have distinct weights. In this paper, we determine the gracefulness of the union of cycles and complete bipartite graphs. We also give graceful labelings of unions of complete bipartite graphs.

The expected value and the variance of a multiplicity of a given part size in a random composition of an integer is obtained. This result was used in [1] to analyze algorithms for computing the Walsh-Hadamard transform.

We enumerate the balanced tournament designs on 10 points (BTD(5)) and find that there are exactly 30,220,557 nonisomorphic designs. We also find that there are exactly two nonisomorphic partitioned BTD(5)’s and 8,081,114 factored BTD(5)’s on 10 points. We enumerate other classes of balanced tournament designs on 10 points and give examples of some of the more interesting ones. In 1988, Corriveau enumerated the nonisomorphic BTD(4)’s, finding that there are 47 of them. This paper enumerates the next case and provides another good example of the combinatorial explosion phenomenon.

We examine decompositions of complete graphs with an even number of vertices into isomorphic spanning trees. We develop a cyclic factorization of \( K_{2n} \) into non-symmetric spanning trees. Our factorization methods are based on flexible \( q \)-labeling and blended labeling, introduced by Froncek. In this paper, we present several infinite classes of non-symmetric trees which have flexible \( q \)-labeling or blended labeling.

The analysis of the Tutte polynomial of a matroid using activities is associated with a shelling of the family of spanning sets. We introduce an activities analysis of the reliability of a system specified by an arbitrary clutter, associated with an \( \mathcal{S} \)-partition rather than a shelling. These activities are related to a method of constructing Boolean interval partitions developed by Dawson in the early 1980s.

Let \(D\) be a connected symmetric digraph, \(\Gamma\) a group of automorphisms of \(D\), and \(A\) a finite abelian group. For a cyclic \(A\)-cover of \(D\), we consider a lift of \(\gamma \in \Gamma\), and the associated group automorphism of some subgroup of \(Aut \, A\). Furthermore, we give a characterization for any \(\gamma \in \Gamma\) to have a lift in terms of some matrix.