An \(\lambda\)-design on \(v\) points is a set of \(v\) subsets (blocks) of a \(v\)-set such that any two distinct blocks meet in exactly \(\lambda\) points and not all of the blocks have the same size. Ryser’s and Woodall’s \(\lambda\)-design conjecture states that all \(\alpha\)-designs can be obtained from symmetric designs by a complementation procedure. In a previous paper, the author established feasibility criteria for the existence of \(\lambda\)-designs with two block sizes in the form of integrality conditions, equations, inequalities, and Diophantine equations involving various parameters of the designs. In that paper, these criteria and a computer were used to prove that the \(\lambda\)-design conjecture is true for all \(\lambda\)-designs with two block sizes with \(\lambda \leq 90\) and \(\lambda \neq 45\). In this paper, we extend these results and prove that the \(\lambda\)-design conjecture is also true for all \(\lambda\)-designs with two block sizes with \(\lambda = 45\) or \(91 \leq \alpha < 150\).

The binary linear code \(H^\bot_{m,2}\), \(m > 2\), of length \(\binom{m}{2}\) represented by the generator matrix \(H_{m,2}\) consisting of all distinct column strings of length \(m\) and Hamming weight \(2\) is considered. A parity-check matrix \(H^\bot_{m,2}\) is assigned to the code \(H^\bot_{m,2}\). The code \(H_{m,2,3}\), \(m > 3\), of length \(\binom{m}{2} + \binom{m}{3}\) represented by the parity-check matrix \(H_{m,2,3}\) consisting of all distinct column strings of length \(m\) and Hamming weight two or three is also considered. It is shown that \(H^\bot_{m,2}\) and \(H_{m,2,3}\) are optimal stopping redundancy codes, that is for each of these codes the stopping distance of the associated parity-check matrix is equal to the minimum Hamming distance of the code, and the rows of the parity-check matrix are linearly independent. Explicit formulas determining the number of stopping sets of arbitrary size for these codes are given.

For a finite group \(G\) and subsets \(T_1, T_2\) of \(G\), the Bi-Cayley digraph \(D = (V(D), E(D)) = D(G, T_1, T_2)\) of \(G\) with respect to \(T_1\) and \(T_2\) is defined as the bipartite digraph with vertex set \(V(D) = G \times \{0, 1\}\), and for \(g_1, g_2 \in G\), \(((g_1, 0), (g_2, 1)) \in E(D)\) if and only if \(g_2 = t_1 g_1\) for some \(t_1 \in T_1\), and \(((g_1, 1), (g_2, 0)) \in E(D)\) if and only if \(g_1 = t_2 g_2\) for some \(t_2 \in T_2\). If \(|T_1| = |T_2| = k\), then \(D\) is \(k\)-regular. In this paper, the spectra of Bi-Circulant digraphs are determined. In addition, some asymptotic enumeration theorems for the number of directed spanning trees in Bi-Circulant digraphs are presented.

The genus of a graph \(G\), denoted by \(\gamma(G)\), is the minimum genus of an orientable surface in which the graph can be embedded. In the paper, we use the Joint Tree Model to immerse a graph on the plane and obtain an associated polygon of the graph. Along the way, we construct a genus embedding of the edge disjoint union of \(K\) and \(H\), and solve Michael Stiebitz’s proposed conjecture: Let \(G\) be the edge disjoint union of a complete graph \(K\) and an arbitrary graph \(H\). Let \(H’\) be the graph obtained from \(H\) by contracting the set \(V(X)\) to a single vertex. Then

\[\gamma(K) + \gamma(H’) \leq \gamma(G).\]

We investigate brother avoiding round robin doubles tournaments and construct several infinite families. We show that there is a BARRDT(\(x\)) that is not a SAMDRR(\(n\)) for all \(n > 4\).

A digraph \(D(V, E)\) is said to be graceful if there exists an injection \(f: V(G) \to \{0, 1, \ldots, |E|\}\) such that the induced function \(f’: E(G) \to \{1, 2, \ldots, |E|\}\) which is defined by \(f'(u, v) = [f(v) – f(u)] \pmod{|E| + 1}\) for every directed edge \((u, v)\) is a bijection. Here, \(f\) is called a graceful labeling (graceful numbering) of \(D(V, E)\), while \(f’\) is called the induced edge’s graceful labeling of \(D\). In this paper, we discuss the gracefulness of the digraph \(n – \overrightarrow{C}_m\), and prove that \(n – \overrightarrow{C}_m\) is a graceful digraph for \(m = 4, 6, 8, 10\) and even \(n\).

In this note, we consider relative difference sets with the parameter \((m, 2, m-1, \frac{m-2}{2})\) in a group \(G\) relative to a subgroup \(N\). In the splitting case, \(G = H \times N\), we give a lower bound for the size of the commutator group \(H’\), and we show that \(H\) cannot have a homomorphic image which is generalized dihedral. In the non-splitting case, we prove that there is no \((2n, 2, 2n-1, n-1)\) relative difference set in a generalized dihedral group of order \(4n\), \(n > 1\).

Let \(P_n\) be a path with \(n\) vertices. \(P_n^k\), the \(k\)-th power of the path \(P_n\), is a graph on the same vertex set as \(P_n\), and the edges that join all vertices \(x\) and \(y\) if and only if the distance between them is at most \(k\). In this paper, the crossing numbers of \(P_n^k\) are studied. Drawings of \(P_n^k\) are presented and proved to be optimal for the case \(n \leq 8\) and for the case \(k \leq 4\).

A graph is said to be locally grid if the structure around each of its vertices is a \(3 x 3\) grid. As a follow up of the research initiated in \([8]\) and \([9]\) we prove that most locally grid graphs are uniquely determined by their Tutte polynomial.

Let \(P(G, \lambda)\) be the chromatic polynomial of a graph \(G\). A graph \(G\) is chromatically unique if for any graph \(H\), \(P(H, \lambda) = P(G, \lambda)\) implies \(H\) is isomorphic to \(G\). In his Ph.D. thesis, Zhao [Theorems 5.4.2 and 5.4.3] proved that for any positive integer \(t \geq 3\), the complete \(t\)-partite graphs \(K(p – k, p, p, \ldots, p)\) with \(p \geq k+2 \geq 4\) and \(K(p-k, p – 1, p, \ldots, p)\) with \(p \geq 2k \geq 4\) are chromatically unique. In this paper, by expanding the technique employed by Zhao, we prove that the complete \(t\)-partite graph \(K(p-k,\underbrace{ p -1, \ldots, p-1}, \underbrace{p, \ldots, p})\) is chromatically unique for integers \(p \geq k+2 \geq 4\) and \(t \geq d+3 \geq 3\).