In this paper, we construct two series of balanced incomplete block (BIB) designs with parameters:
\[v = \binom{2m-3}{2} ,r= \frac{(2m-5)!}{(m-1)!}, k= {m}\]
\[b=\frac{(2m-3)!}{2m(m-1)!} , \lambda = \frac{(2m-6)!}{(m-3)!}\]
and
\[v = \binom{2m+1}{2} , b = b_1(m+1), r = 2m(\overline{\lambda}_1-\overline{\lambda}_2), k = m^2\]
\[\lambda = (m-1)(\overline{\lambda}_1-2\overline{\lambda}_2+\overline{\lambda}_3)+m(\overline{\lambda}_2-\overline{\lambda}_3)\]
where \(k_1, b_1, \overline{\lambda}_i\) are parameters of a special \(4-(v, k, \lambda)\) design.

The strong chromatic index of a graph \(G\), denoted \(sq(G)\), is the minimum number of parts needed to partition the edges of \(G\) into induced matchings. The subset graph \(B_m(k)\) is the bipartite graph whose vertices represent the elements and the \(k\)-subsets of an \(m\) element ground set where two vertices are adjacent if and only if the vertices are distinct and the element corresponding to one vertex is contained in the subset corresponding to the other. We show that \(sq(B_m(k)) =\binom{m}{k-1}\) and that this satisfies the strong chromatic index conjecture by Brualdi and Quinn \([3]\) for bipartite graphs.

For a graph \(G\), if \(F\) is a nonempty subset of the edge set \(E(G)\), then the subgraph of \(G\) whose vertex set is the set of ends of edges in \(F\) is denoted by \(_G\). Let \(E(G) = \cup_{i \in I} E_i\) be a partition of \(E(G)\), let \(D_i = _G\) for each \(i\), and let \(\phi = (D_i | i \in I)\), then \(\phi\) is called a partition of \(G\) and \(E_i\) (or \(D_i\)) is an element of \(\phi\). Given a partition \(\phi = (D_i | i \in I)\) of \(G\), \(\phi\) is an admissible partition of \(G\) if for any vertex \(v \in V(G)\), there is a unique element \(D_i\) that contains vertex \(v\) as an inner point. For two distinct vertices \(u\) and \(v\), a \(u-v\) walk of \(G\) is a finite, alternating sequence \(u = u_0, e_1, u_1, e_2, \ldots, v_{n.1},e_n,u_n = v\) of vertices and edges, beginning with vertex \(u\) and ending with vertex \(v\), such that \(e_i = u_{i-1}u_i\) for \(i = 1, 2, \ldots, n\). A \(u-v\) string is a \(u-v\) walk such that no vertex is repeated except possibly \(u\) and \(v\), i.e., \(u\) and \(v\) are allowed to appear at most two times. Given an admissible partition \(\phi\), \(\phi\) is a string decomposition or \(SD\) of \(G\) if every element of \(\phi\) is a string.

In this paper, we prove that a \(2\)-connected graph \(G\) has an \(SD\) if and only if \(G\) is not a cycle. We also give a characterization of the graphs with cut vertices such that each graph has an \(SD\).

The cyclic chromatic number is the smallest number of colours needed to colour the nodes of a tournament so that no cyclic triple is monochromatic. Bagga, Beineke, and Harary \({[1]}\) conjectured that every tournament score vector belongs to a tournament with cyclic chromatic number \(1\) or \(2\). In this paper, we prove this conjecture and derive some other results.

A path of a graph is maximal if it is not a proper subpath of any other path of the graph. The path spectrum is the set of lengths of all maximal paths in the graph. A graph is scenic if its path spectrum is a singleton set. In this paper, we give a new proof characterizing all scenic graphs with a Hamiltonian path; this result was first proven by Thomassen in \(1974\). The proof strategy used here is also applied in a companion paper in which we characterize scenic graphs with no Hamiltonian path.