In this paper, we consider the problem of determining the structure of a minimal critical set in a latin square \(L\) representing the elementary abelian \(2\)-group of order \(8\).

In this paper, the first two (resp. four) largest signless Laplacian spectral radii together with the corresponding graphs in the class of bicyclic (resp. tricyclic) graphs of order n are determined, and the first two (resp. four) largest signless Laplacian spreads together with the corresponding graphs in the class of bicyclic (resp. tricyclic) graphs of order \(n\) are identified.

An edge-magic total labeling of a graph \(G\) is a one-to-one map \(\lambda\) from \(V(G) \cup E(G)\) onto the integers \(\{1, 2, \ldots, |V(G) \cup E(G)|\}\) with the property that there exists an integer constant \(c\) such that \(\lambda(x) + \lambda(x,y) + \lambda(y) = c\) for any \((x, y) \in E(G)\). If \(\lambda(V(G)) = \{1, 2, \ldots, |V(G)|\}\), then the edge-magic total labeling is called super edge-magic total labeling. In this paper, we formulate super edge-magic total labeling on subdivisions of stars \(K_{1,p}\), for \(p \geq 5\).

In this paper, we briefly survey Euler’s works on identities connected with his famous Pentagonal Number Theorem. We state a partial generalization of his theorem for partitions with no part exceeding an identified value \(k\), along with some identities linking total partitions to partitions with distinct parts under the above constraint. We derive both recurrence formulas and explicit forms for \(\Delta_n(m)\), where \(\Delta_n(m)\) denotes the number of partitions of \(m\) into an even number of distinct parts not exceeding \(n\), minus the number of partitions of \(m\) into an odd number of distinct parts not exceeding \(n\). In fact, Euler’s Pentagonal Number Theorem asserts that for \(m \leq n\), \(\Delta_n(m) = \pm 1\) if \(m\) is a Pentagonal Number and \(0\) otherwise. Finally, we establish two identities concerning the sum of bounded partitions and their connection to prime factors of the bound integer.

Consider the following one-person game: let \(S = {F_1, F_2,\ldots, F_r}\) be a family of forbidden graphs. The edges of a complete graph are randomly shown to the Painter one by one, and he must color each edge with one of \(r\) colors when it is presented, without creating some fixed monochromatic forbidden graph \(F\); in the \(i\)-th color. The case of all graphs \(F\); being cycles is studied in this paper. We give a lower bound on the threshold function for online \(S\)-avoidance game,which generalizes the results of Marciniszyn, Spdhel and Steger for the symmetric case. [Combinatorics, Probability and Computing, Vol. \(18, 2009: 271-300.\)]

Given positive integers \(n\), \(k\), and \(m\), the \((n,k)\)-th \(m\)-restrained Stirling number of the first kind is the number of permutations of an \(n\)-set with \(k\) disjoint cycles of length \(\leq m\). By inverting the matrix consisting of the \((n,k)\)-th \(m\)-restrained Stirling number of the first kind as the \((n+1,k+1)\)-th entry, the \((n,k)\)-th \(m\)-restrained Stirling number of the second kind is defined. In this paper, we study the multi-restrained Stirling numbers of the first and second kinds to derive their explicit formulae, recurrence relations, and generating functions. Additionally, we introduce a unique expansion of multi-restrained Stirling numbers for all integers \(n\) and \(k\), and a new generating function for the Stirling numbers of the first kind.

Employing \(q\)-commutive structures, we develop binomial analysis and combinatorial applications induced by an important operator in
analogue Fourier analysis associated with well-known \(q\)-series of L.J. Rogers.

In [H. Ngo, D. Du, New constructions of non-adaptive and error-tolerance
pooling designs, Discrete Math. \(243 (2002) 167-170\)], by using subspaces
in a vector space Ngo and Du constructed a family of well-known pooling
designs. In this paper, we construct a family of pooling designs by using
bilinear forms on subspaces in a vector space, and show that our design and
Ngo-Du’s design have the same error-tolerance capability but our design is
more economical than Ngo-Du’s design under some conditions.

A transverse Steiner quadruple system \((TSQS)\) is a triple \((X, \mathcal{H}, \mathcal{B})\) where \(X\) is a \(v\)-element set of points, \(\mathcal{H} = \{H_1, H_2, \ldots, H_r\}\) is a partition of \(X\) into holes, and \(\mathcal{B}\) is a collection of transverse \(4\)-element subsets with respect to \(\mathcal{H}\), called blocks, such that every transverse \(3\)-element subset is in exactly one block. In this article, we study transverse Steiner quadruple systems with \(r\) holes of size \(g\) and \(1\) hole of size \(u\). Constructions based on the use of \(s\)-fans are given, including a construction for quadrupling the number of holes of size \(g\). New results on systems with \(6\) and \(11\) holes are obtained, and constructions for \(\text{TSQS}(x^n(2n)^1)\) and \(\text{TSQS}(4^n2^1)\) are provided.

Let \(G\) be a subgraph of the complete graph \(K_{r+1}\) on \(r+1\) vertices, and let \(K_{r+1} – E(G)\) be the graph obtained from \(K_{r+1}\) by deleting all edges of \(G\). A non-increasing sequence \(\pi = (d_1, d_2, \ldots, d_n)\) of nonnegative integers is said to be potentially \(K_{r+1} – E(G)\)-graphic if it is realizable by a graph on \(n\) vertices containing \(K_{r+1} – E(G)\) as a subgraph. In this paper, we give characterizations for \(\pi = (d_1, d_2, \ldots, d_n)\) to be potentially \(K_{r+1} – E(G)\)-graphic for \(G = 3K_2, K_3, P_3, K_{1,3}\), and \(K_2 \cup P_2\), which are analogous to Erdős-Gallai’s characterization using a system of inequalities. These characterizations partially answer one problem due to Lai and Hu [10].