In this note, we give a characterization of regular graphs which are neutral.

It is known that a pair of mutually orthogonal Latin squares (MOLS) of order \(n\) can be embedded in a pair of MOLS of order \(t\) if \(t \geq 3n\). Here, we discuss the prospects of extending this result to the case when the smaller pair is only a pair of mutually orthogonal \({partial}\) Latin squares (MOPLS). We obtain some conditions, analogous to those of Ryser for embedding partial Latin squares in complete Latin squares, which we show are necessary for the embedding of MOPLS. We discuss also some implications if these conditions are, in fact, also sufficient.

We also discuss the analogous problem for pairs of partial Kirkman triple systems (PKTS).

If the non-zero entries of an incidence matrix \(X\) of BIBD\((v, b, r, k, 2)\) have been signed to produce a \((0, 1, -1)\) matrix \(Y\0 such that

\[YY^T = rI_v,\]

then we say it has been signed. The resulting matrix \(Y\) is said to be a Bhaskar Rao design BRD\((v, k, 2)\). We discuss the complexity of two signing problems, (i) Given \(v\), \(k\), and \(\lambda\), decide whether there is a BRD\((v, k, 2\lambda)\), (ii) Given a BIBD\((v, k, 2\lambda)\), decide whether it is signable.
The paper describes related optimisation problems. We show that the signing problems are equivalent to finding the real roots of certain multi-variable polynomials. Then we describe some linear constraints which reduce the size of the second problem, we show certain special cases have polynomial complexity, and we indicate how in some cases the second problem can be decomposed into smaller independent problems. Finally, we characterise signable Steiner triple systems in terms of their block-intersection graphs, and show that the complexity of deciding whether a twofold triple system can be signed is linear in the number of blocks.

Four \((1, -1, 0)\)-matrices of order \(m\), \(X = (x_{ij})\), \(Y = (y_{ij})\), \(Z = (z_{ij})\), \(U = (u_{ij})\) satisfying

1. \(XX^T + YY^T + ZZ^T + UU^T= 2mI_m\),
2. \(x_{ij}^2 + y_{ij}^2 + z_{ij}^2 + u_{ij}^2 = 2, i,j = 1,\ldots,m\),
3. \(X, Y, Z, U\) mutually amicable,

will be called semi Williamson type matrices of order \(m\). In this paper, we prove that if there exist Williamson type matrices of order \(n_1, \ldots, n_k\), then there exist semi Williamson type matrices of order \(N = \prod_{j=1}^k n_j^{r_j}\), where \(r_j\) are non-negative integers. As an application, we obtain a \(W(4N, 2N)\).
Although the paper presents no new \(W(4n, 2n)\) for \(n\) odd, \(n < 3000\), it is a step towards proving the conjecture that there exists a \(W(4n, 2n)\) for any positive integer \(n\). This conjecture is a sub-conjecture of the Seberry conjecture \([4, page 92]\) that \(W(4n, k)\) exist for all \(k = 0, 1, \ldots, 4n\). In addition, we find infinitely many new \(W(2n, n)\), \(n\) odd and the sum of two squares.

A multi-set design of order \(v\), \({MB}_t(v, k, \lambda)\), first defined by Assaf, Hartman, and Mendelsohn, is an ordered pair \((V, B)\), where \(V\) is a set of cardinality \(v\) and \(B\) is a collection of multi-subsets of \(V\) of size \(k\) (called blocks), with the property that every multi-subset of \(V\) of size \(t\) is contained a total of \(\lambda\) times in the blocks of \(B\). (For example, the multi-set \(\{a,b,b\}\) is contained \(\binom{2}{1}\binom{4}{2} = 12\) times in the multi-set \(\{a,a,b,b,b,b,c\}\) and not at all in the multi-set \(\{a,a,b,c\}\).) Previously, the first author had pointed out that any \(t\)-multi-set design is a \(1\)-design. Here, we show the pleasant yet not obvious fact that any \(t\)-multi-set design is a \(t’\)-multi-set design for any positive integer \(t’ \leq t\).

In this paper, we introduce the concept of node expansion. Node expansion is a generalization of edge subdivision and an inverse of subgraph contraction. A graph \(G_1 = (V_1, E_1)\) is an \(H\)-node expansion of \(G = (V, E)\) if and only if \(G_1\) contains a subgraph \(H = (V_H, E_H)\) such that \(V = V_1 – V_H \cup \{v\}\) and \(E = E_1 – E_H \cup \{vw | wh \in E_1 \;\text{and}\; h \in V_H\} \cup \{v\}\). The concept of node expansion is of considerable importance in modernization of networks.

We consider the node expansion problem of transforming a graph to a bipartite graph with a minimum number of node expansions using \(K_2\). We show that the \(K_2\)-node expansion problem is NP-Complete for general graphs and remains so if the input graph has maximum degree 3. However, we present a \(O({n}^2 \log n + mn + p^3)\) algorithm for the case when the input graph is restricted to be planar \(3\)-connected and output graph is required to be planar bipartite.

An algorithm is presented for finding all \((0,1)\)-solutions to the matrix problem \(AX = J\), where \(A\) is a \((0,1)\)-matrix and \(J\) is the all \(1\)’s column vector. It is applied to the problem of enumerating distinct cyclic Steiner systems and five new values are obtained. Specifically, the number of distinct solutions to \(S(2,3,55), S(2,3,57), S(2,3,61), S(2,3,63)\), and \(S(3,4,22)\) are \(121,098,240, 84,672,512, 2,542,203,904, 1,782,918,144\), and \(1140\), respectively.

Let \(g_k(n) = \sum_{\underline{v}\in C_k(n)} \binom{n}{v} 2^{v_1v_2 + v_2v_3 + v_3v_4 + \ldots +v_{k-1}v_k}\) where \(C_k(n)\) denote the set of \(k\)-compositions of \(n\). We show that

1. \(g_k(n+p-1) \equiv g_k(n) \pmod{p}\) for all \(k,n \geq 1\), prime \(p\);
2. \(g_k(n)\) is a polynomial in \(k\) of degree \(n\) for \(k \geq n+1\);

and, moreover, that these properties hold for wider classes of functions which are sums involving multinomial coefficients.