In the spirit of Ryser’s theorem, we prove sufficient conditions on \(k\), \(\ell\), and \(m\) so that \(k \times \ell \times m\) Latin boxes, i.e., partial Latin cubes whose filled cells form a \(k \times \ell \times m\) rectangular box, can be extended to a \(k \times n \times m\) Latin box, and also to a \(k \times n \times m\) Latin box, where \(n\) is the number of symbols used, and likewise the order of the Latin cube. We also prove a partial Evans-type result for Latin cubes, namely that any partial Latin cube of order \(n\) with at most \(n-1\) filled cells is completable, given certain conditions on the spatial distribution of the filled cells.

A star-factor of a graph \(G\) is a spanning subgraph of \(G\) such that each component is a star. An edge-weighting of \(G\) is a function \(w: E(G) \rightarrow \mathbb{N}^+\), where \(\mathbb{N}^+\) is the set of positive integers. Let \(\Omega\) be the family of all graphs \(G\) such that every star-factor of \(G\) has the same weight under some fixed edge-weighting \(w\). The open problem of characterizing the class \(\Omega\), posed by Hartnell and Rall, is motivated by the minimum cost spanning tree and the optimal assignment problems. In this paper, we present a simple structural characterization of the graphs in \(\Omega\) that have girth at least five.

We show that whenever the length four words over a three letter alphabet are two-colored, there must exist a monochromatic combinatorial line. We also provide some computer generated lower bounds for some other Hales-Jewett numbers.

This paper introduces a method for finding closed forms for certain sums involving squares of binomial coefficients. We use this method to present an alternative approach to a problem of evaluating a different type of sums containing squares of the numbers from
Catalan’s triangle.

In this paper, we deal with a special kind of hypergraph decomposition. We show that there exists a decomposition of the 3-uniform hypergraph \(\lambda K_v^{(3)}\) into a special kind of hypergraph \(K_{4}^{(3)} – e\) whose leave has at most two edges, for any positive integers \(v \geq 4 \) and \(\lambda\).

For a connected graph \(G\) of order \(n \geq 2\) and a linear ordering \(s = v_1, v_2, \ldots, v_n\) of \(V(G)\), define \(d(s) = \sum_{i=1}^{n-1} d(v_i, v_{i+1})\), where \(d(v_i, v_{i+1})\) is the distance between \(v_i\) and \(v_{i+1}\). The traceable number \(t(G)\) and upper traceable number \(t^+(G)\) of \(G\) are defined by \(t(G) = \min\{d(s)\}\) and \(t^+(G) = \max\{d(s)\}\), respectively, where the minimum and maximum are taken over all linear orderings \(s\) of \(V(G)\). The traceable number \(t(v)\) of a vertex \(v\) in \(G\) is defined by \(t(v) = \min\{d(s)\}\), where the minimum is taken over all linear orderings \(s\) of \(V(G)\) whose first term is \(v\). The \({maximum\; traceable \;number}\) \(t^*(G)\) of \(G\) is then defined by \(t^*(G) = \max\{t(v) : v \in V(G)\}\). Therefore, \(t(G) \leq t^*(G) \leq t^+(G)\) for every nontrivial connected graph \(G\). We show that \(t^*(G) \leq \lfloor \frac{t(G)+t^+(G)+1}{2}\rfloor\) for every nontrivial connected graph \(G\) and that this bound is sharp. Furthermore, it is shown that for positive integers \(a\) and \(b\), there exists a nontrivial connected graph \(G\) with \(t(G) = a\) and \(t^*(G) = b\) if and only if \(a \leq b \leq \left\lfloor \frac{3n}{2} \right\rfloor\).

Let \(G\) be a simple graph with \(n\) vertices and \(m\) edges, and let \(\lambda_1\) and \(\lambda_2\) denote the largest and second largest eigenvalues of \(G\). For a nontrivial bipartite graph \(G\), we prove that:
(i) \(\lambda_1 \leq \sqrt{m – \frac{3-\sqrt{5}}{2}}\), where equality holds if and only if \(G \cong P_4\);
(ii) If \(G \ncong P_n\), then \(\lambda_1 \leq \sqrt{{m} – (\frac{5-\sqrt{17}}{2})}\), where equality holds if and only if \(G \cong K_{3,3} – e\);
(iii) If \(G\) is connected, then \(\lambda_2 \leq \sqrt{{m} – 4{\cos}^2(\frac{\pi}{n+1})}\), where equality holds if and only if \(G \cong P_{n,2} \leq n \leq 5\);
(iv) \(\lambda_2 \geq \frac{\sqrt{5}-1}{2}\), where equality holds if and only if \(G \cong P_4\);
(v) If \(G\) is connected and \(G \ncong P_n\), then \(\lambda_2 \geq \frac{5-\sqrt{17}}{2}\), where equality holds if and only if \(G \cong K_{3,3} – e\).

Let \(n\) be a positive integer. Denote by \(PG(n,q)\) the \(n\)-dimensional projective space over the finite field \(\mathbb{F}_q\) of order \(q\). A blocking set in \(PG(n,q)\) is a set of points that has non-empty intersection with every hyperplane of \(PG(n,q)\). A blocking set is called minimal if none of its proper subsets are blocking sets. In this note, we prove that if \(PG(n_i,q)\) contains a minimal blocking set of size \(k_i\) for \(i \in \{1,2\}\), then \(PG(n_1 + n_2 + 1,q)\) contains a minimal blocking set of size \(k_1 + k_2 – 1\). This result is proved by a result on groups with maximal irredundant covers.

A graph is said to be edge-transitive if its automorphism group acts transitively on its edge set. In this paper, all connected cubic edge-transitive graphs of order \(12p\) or \(12p^2\) are classified.

For any \(n\geq 7\), we prove that there exists a tournament of order \(n\), such that for each pair of distinct vertices there exists a path of length \(2\).