In this paper, we give a complete solution to the existence of lattice group divisible \(3\)-designs with block sizes four and six.

Let \( R \) be a commutative ring with identity and \( \mathbb{A}^*(R) \) be the set of non-zero ideals with non-zero annihilators. The annihilating-ideal graph of \( R \) is defined as the graph \( \mathbb{AG}(R) \) with the vertex set \( \mathbb{A}^*(R) \) and two distinct vertices \( I_1 \) and \( I_2 \) are adjacent if and only if \( I_1 I_2 = (0) \). In this paper, we study some connections between the graph-theoretic properties of \( \mathbb{AG}(R) \) and algebraic properties of the commutative ring \( R \).

Let \( A_n = (a_1, a_2, \ldots, a_n) \) and \( B_n = (b_1, b_2, \ldots, b_n) \) be two sequences of nonnegative integers satisfying \( a_1 \geq a_2 \geq \cdots \geq a_n \), \( a_i \leq b_i \) for \( i = 1,2,\ldots,n \), and \( a_i = a_{i+1} \) implies that \( b_i \geq b_{i+1} \) for \( i = 1,2,\ldots,n-1 \). Let \( I \) be a subset of \( \{1,2,\ldots,n\} \) and \( a_i \equiv b_i \pmod{2} \) for each \( i \in I \). \( (A_n; B_n) \) is said to be partial parity graphic with respect to \( I \) if there exists a simple graph \( G \) with vertices \( v_1, v_2, \ldots, v_n \), such that \( a_i \leq d_G(v_i) \leq b_i \) for \( i = 1,2,\ldots,n \) and \( d_G(v_i) \equiv b_i \pmod{2} \) for each \( i \in I \). In this paper, we give a characterization for \( (A_n; B_n) \) to be partial parity graphic. This is a variation of the partial parity \( (g, f) \)-factor theorem due to Kano and Matsuda in degree sequences.

It’s well known that all of the pooling designs constructed are based on a finite set or a finite vector space. In this paper, we construct two families of pooling designs not only based on finite sets (resp. finite vector spaces) but also on partial mappings (resp. partial linear mappings), and discuss their error-tolerance properties.

Let \( 2^{[m]} \) be ordered by set inclusion, and let \( \mathcal{B} \subseteq 2^{[m]} \) be an antichain. An antichain \( \mathcal{B} \) is called \( k \)-regular (\( k \in \mathbb{N} \)) if for each \( i \in [m] \) there are exactly \( k \) blocks \( B_1, B_2, \ldots, B_k \in \mathcal{B} \) containing \( i \). An antichain is called flat if there exists a positive integer \( l \) such that \( l \leq |B| \leq l+1 \) for all \( B \in \mathcal{B} \), and we call an antichain maximal if the collection of sets \( \mathcal{B} \cup \{B\} \) is not an antichain for all \( B \notin \mathcal{B} \). We call a maximal \( k \)-regular antichain \( \mathcal{B} \subseteq \binom{[m]}{2} \cup \binom{[m]}{3} \) a \( (k,m) \)-MFRAC. In this paper we analyze \( (k,m) \)-MFRACs in the cases \( m \leq 7 \), \( k = m \), \( k = m-1 \), and \( k = m-2 \). We provide some constructions, give necessary conditions for existence, and mention some open problems.

Generalized binomial coefficients are considered. The aim of this paper is to provide a new general combinatorial interpretation of the Lucas-nomial and \( (p,q) \)-nomial coefficients in terms of tiling of \( d \)-dimensional rectangular boxes. The recurrence relation of these numbers is proved in a combinatorial way. To this end, our results are extended to the case of corresponding multi-nomial coefficients.

In this paper, we determine the necessary and sufficient conditions for the existence of simple incomplete triple systems for all \( \lambda \leq 6 \).

Dudeney’s round table problem asks for a set of Hamilton cycles in \( K_n \), having the property that each \( 2 \)-path in \( K_n \) lies in exactly one of the cycles. In this paper, we show how to construct a solution of Dudeney’s round table problem for even \( n \) from a semi-antipodal Hamilton decomposition of \( K_{n-1} \).

Using the spectral invariants of graphs, we present sufficient conditions for some stable properties of graphs.

A matching \( M \) in a graph \( G \) is a subset of \( E(G) \) in which no two edges have a vertex in common. A vertex \( V \) is unsaturated by \( M \) if there is no edge of \( M \) incident with \( V \). A matching \( M \) is called a perfect matching if there is no vertex of the graph that is unsaturated by \( M \). Let \( G \) be a \( k \)-edge-connected graph, \( k \geq 1 \), on even \( n \) vertices, with minimum degree \( r \) and maximum degree \( r + e \), \( e \geq 1 \). In this paper, we find a lower bound for \( n \) when \( G \) has no perfect matchings.