It is known that the ovoids in \({O}_5(q)\), \(q \leq 7\), are classical ovoids. Using algebraic and computational techniques, we classify ovoids in \({O}_5(9)\) and \({O}_5(11)\) with the aid of a computer. We also study the ovoids which contain an irreducible conic and classify them in \({O}_5(13)\). Our results show that there is only one nonclassical ovoid (a member from a family of Kantor) up to isomorphism in \({O}_5(9)\) and all the ovoids in \({O}_5(11)\) are classical.

A symmetric design \((U, \mathcal{A})\) is a strong subdesign of a symmetric design \((V, \mathcal{B})\) if \(U \subseteq V\) and \(\mathcal{A}\) is the set of non-empty intersections \(B \cap U\), where \(B \in \mathcal{B}\). We demonstrate three constructions of symmetric designs, where this notion is useful, and produce two new infinite families of symmetric designs with parameters \(v = \left(\frac{73^{m+1} – 64}{9}\right), k = 73^m,\lambda = 9 \cdot 73^{m-1}\) and \(v = 1+2(q + 1)\left(\frac{(q + 1)^{2m} – 1}{q+2}\right), k = (q + 1)^{2m}, \lambda = \frac{(q + 1)^{2m-1} (q + 2)}{2}\) where \(m\) is a positive integer and \(q = 2^p – 1\) is a Mersenne prime. The main tools in these constructions are generalized Hadamard matrices and balanced generalized weighing matrices.

The least deviant path was defined by Klostermeyer \([1]\) as the path between two vertices \(u\) and \(v\) that minimizes the difference between the largest and smallest weights on the path. This paper presents an \(O(E \log E)\) time algorithm for this problem in undirected graphs, improving upon the previously given \(O(E^{1.793})\) time algorithm.
The same algorithm can also be used to solve the problem in \(O(VE)\) time in directed graphs.

It is well-known that the set of all circulations of a connected digraph \(G\) on \(p\) vertices with \(q\) edges forms a ternary linear code \(\text{C} = \text{C}_E(G)\)
with parameters \([q, q – p + 1, g]\), where \(g\) is the girth of \(G\). Such codes were first studied by Hakimi and Bredeson \([8]\) in \(1969\), who investigated problems
of augmenting \(\text{C}\) to a larger \((q, k, g)\)-code and efficiently decoding such codes. Their treatment was similar to their previous work on binary codes \([4, 7]\).
Recently, we have made significant progress in the binary case by generalizing Hakimi’s and Bredeson’s construction method to obtain better augmenting codes and developing a more efficient decoding algorithm. In this paper, we explore how our methods can be
adapted to achieve corresponding progress in the ternary case. In particular, we will correct an oversight in a graph-theoretic lemma of Bredeson and Hakimi, which affects their decoding algorithms and discuss alternative decoding procedures based on a connection to a challenging optimization problem.

Let \(G\) be a graph and let \(S\) be a subset of vertices of \(G\). The open neighborhood of a vertex \(v\) of \(G\) is the set of all vertices adjacent to \(v\) in \(G\). The set \(S\) is an open packing of \(G\) if the open neighborhoods of the vertices of \(S\) are pairwise disjoint in \(G\). The lower open packing number of \(G\), denoted \(\rho_L^o(G)\), is the minimum cardinality of a maximal open packing of \(G\), while the (upper) open packing number of \(G\), denoted \(\rho^o(G)\), is the maximum cardinality among all open packings of \(G\). In this paper, we present theoretical and computational
results for the open packing numbers of a graph.

The two-color Ramsey number \(R(k, l)\) is the smallest integer \(p\) such that for any graph \(G\) on \(p\) vertices either \(G\) contains a \(K_k\) or \(\overline{G}\) contains a \(K_l\), where \(\overline{G}\) denotes the complement of \(G\). A new upper bound formula is given for two-color Ramsey numbers. For example, we get \(R(7,9) \leq 1713\),
\(R(8,10) \leq 6090\) etc.

Let \(M\) be a finite dimensional commutative nilpotent algebra over a field \(K\) of prime characteristic \(p\). It has been conjectured that \(\dim M \geq p \;\dim M^{(p)}\), where \(M^{(p)}\) is the subalgebra of \(M\) generated by \(x^p\), \(x \in M\), \([2]\).This was proved (by Eggert) in the case \(\dim M^{(p)} \leq 2\) in \(1971\). This result was extended to the noncommutative case in \(1994\) \([8]\). Not only is this conjecture important in its own right, but it was shown (by Eggert) that a proof of the above conjecture would result in a complete classification of the group of units of finite commutative rings of characteristic \(p\) with an identity. In this short paper, we obtain a proof of Eggert’s conjecture in the case \(\dim M^{(p)} = 3\).

Černý, Horák, and Wallis introduced a generalization of Kirkman’s Schoolgirl Problem to the case where the number of schoolgirls is not a multiple of three; they require all blocks to be of size three, except that each resolution class should contain either one block of size two (when \(v \equiv 2 \pmod{3}\)) or one block of size four (when \(v \equiv 1 \pmod{3}\)). We consider the problem of determining the maximum (resp. minimum) possible number of resolution classes such that any pair of elements (schoolgirls) is covered at most (resp. at least) once.

A simple undirected graph \(G\) is called a \emph{sum graph} if there exists a labelling \(\lambda\) of the vertices of \(G\) into distinct positive integers such that any two distinct vertices \(u\) and \(v\) of \(G\) are adjacent if and only if there is a vertex \(w\) whose label \(\lambda(w) = \lambda(u) + \lambda(v)\). It is obvious that every sum graph has at least one isolated vertex, namely the vertex with the largest label. The \emph{sum number} \(\sigma(H)\) of a connected graph \(H\) is the least number \(r\) of isolated vertices \(\overline{K}_r\) such that \(G = H + \overline{K}_r\) is a sum graph.
It is clear that if \(H\) is of size \(m\), then \(\sigma(H) \leq m\). Recently, Hartsfield and Smyth showed that for wheels \(W_n\) of order \(n+1\) and size \(m = 2n\), \(\sigma(W_n) \in \Theta(m)\); that is, that the sum number is of the same order of magnitude as the size of the graph. In this paper, we refine these results to show that for even \(n \geq 4\), \(\sigma(W_n) = {n}/{2} + 2\), while for odd \(n \geq 5\) we disprove a conjecture of Hartsfield and Smyth by showing that \(\sigma(W_n) = n\). Labellings are given that achieve these minima.