In this paper we study group divisible designs (GDDs) with block size 4 and two groups of different sizes when \(\lambda_{2}=1\). We obtain necessary conditions for the existence of such GDDs and prove that these necessary conditions are sufficient in several cases. Further, we present general constructions using resolvable designs.

The Narumi-Katayama index of a graph is defined as the product of the degrees of all vertices in the graph. In this paper, we obtain upper bounds of the Narumi-Katayama index of a connected graph. We further present sufficient conditions based on the Narumi-Katayama index and other graph invariants for Hamiltonian and traceable graphs.

Christoph, Müyesser and Wigderson recently asked whether an approximate form of the Erdős–Sós conjecture is robust under random edge deletions. We establish a density-sensitive transference theorem that converts global resilience for bounded-degree trees in sparse random graphs into resilient forest universality in arbitrary dense host graphs. More precisely, if \(F\) is an \(N\)-vertex graph of edge density \(\lambda\) bounded away from zero and \(p\in[K/N,1]\), then, with probability \(1-o(1)\) uniformly over the host and the percolation parameter, every subgraph obtained from \(F_p\) by deleting at most an \(\alpha\)-fraction of its edges contains every bounded-degree forest on at most \(((1-\alpha)\lambda-\xi)N\) vertices. Consequently, for every \(c>0\), \(L\ge1\), fixed \(D\) and \(\alpha<c\), every graph \(F\) on at most \(Ld\) vertices with average degree at least \(d\) has the property that, after percolation at any rate \(p\in[K/d,1]\) and any subsequent deletion of at most an \(\alpha\)-fraction of the surviving edges, the remaining graph is universal for all forests on at most \((1-c)d\) vertices and maximum degree at most \(D\). This includes vertex-disjoint packings of any prescribed collection of bounded-degree trees of that total order. We further obtain forest-universality results for graphs whose connected components have vertex-cover number at most \(Cd\), and for graphs that can be brought into this form by deleting sufficiently few edges on the \(dn\)-scale.

This paper investigates the combinatorial properties, invariants, and structures arising from the action of the direct product \(S_n \times A_n\) on the Cartesian product \(X^{(2)} \times Y^{(2)}\), where \(X^{(2)}\) and \(Y^{(2)}\) denote the sets of unordered 2-element subsets of two distinct sets \(X\) and \(Y\), each of cardinality \(n\). Using the Orbit-Stabilizer Theorem, we establish that the action is transitive for all \(n \ge 2\) and, through block theory, demonstrate that it is imprimitive for all \(n \ge 3\). By determining the orbits of the stabilizer of a fixed element, we compute the rank of the action to be \(9\) and explicitly enumerate the eight non-trivial subdegrees for \(n \ge 5\) as: \(2(n-2)\), \(2(n-2)\), \(\frac{(n-2)(n-3)}{2}\), \(\frac{(n-2)(n-3)}{2}\), \(4(n-2)^2\), \((n-2)^2(n-3)\), \((n-2)^2(n-3)\), and \(\frac{(n-2)^2(n-3)^2}{4}\). A combinatorial proof establishes that all suborbits are self-paired for \(n\ge 5\). We construct the eight non-trivial suborbital graphs corresponding to these suborbits and analyze their fundamental graph-theoretic properties, including connectedness, regularity, vertex degrees, and girth. The results extend the classical theory of permutation groups acting on combinatorial objects and provide a foundation for potential applications in coding theory, cryptography, and control systems.

For each integer \(m\geq -1\), we study the family of matrices \(\mathsf{S}^{(m)}(n)=\bigl[S(i+j+m, j)\bigr]_{1\leq i, j\leq n},\) whose entries are Stirling numbers of the second kind. We derive several explicit matrix decompositions of \(\mathsf{S}^{(m)}(n)\) in terms of the classical Stirling matrix and certain explicitly constructed upper triangular matrices. As a consequence, we obtain the closed-form determinant formula \(\det \mathsf{S}^{(m)}(n)=\prod_{i=1}^{n} i^{\,i+m},\) which extends several previously known determinant evaluations. We also establish several identities involving both the Stirling numbers of the first and second kinds.

A binary path is a lattice path whose steps are upsteps \(u=(1,1)\) and downsteps \(d=(1,-1)\). A valley is the final point of a descent, that is, of a maximal sequence of consecutive downsteps. We call a valley critical if one of its coordinates is congruent to \(1\) and the other to \(3\) modulo \(4\). We enumerate several classes of binary paths defined by the absence of critical valleys and construct explicit bijections with order-consecutive partition sequences and generalized compositions. We also enumerate binary paths jointly by their length and number of critical valleys and apply these results to the enumeration of Hamiltonian intervals in the lattice of binary paths.

A prime labeling of a simple finite graph \(G\) is a labeling of the vertices with distinct integers from \(\{1,2, \dots, |G|\}\) such that the labels of any two adjacent vertices are coprime. If \(G\) admits such a labeling, we call \(G\) a prime graph. If the set \(\{1,2, \dots, |G|\}\) is replaced by \(\{k,k+1, \dots, k+|G|-1\}\) in the previous definition, then we call it \(k\)-prime labeling, where \(k\) is a positive integer. Since some graph might be 2-prime labeling but not 3-prime labeling, to avoid ambiguity, we define uniform prime labeling for graphs. Focused on trees, using some very classical number theory results, we show that any star is uniform prime if and only if it has at most 16 vertices. We also prove any tree of order \(n\) with diameter \(n-1\), \(n-2\), \(n-3\), and \(n-4\) are uniform prime. Based on these results and several trees with diameter \(n-5\), we show that any tree with at most 8 vertices (There are 1+1+1+2+3+6+11+23=48 such non-isomorphic trees) are uniform prime. We also verified the uniform primality of several periodically constructed trees. We post some open questions and conjectures which should be essential and interesting by the end of Section 4. For example, we conjecture all the trees with order up to 16 are uniform prime and ask if there exists a tree other than star is not uniform prime.

Let \(G(V,E)\) be a finite simple graph. Denote by \(|V|\) and \(|E|\) the cardinality of the set \(V\) and \(E\), respectively. An \(\alpha\)-labeling \(f\) is an injective function \({f}:V\rightarrow \{0,1,2,…, |E|\}\) such that \(\{|f(u)-f(v)|:uv\in E\}=\{1,2,…,|E|\}\) and for a constant \(\lambda\), \(f(u)\le \lambda <f(v)\) for every \(uv\in E\). A graph that has an \(\alpha\)-labeling is called \(\alpha\)-graph. An open chain graph, or shortly a chain graph, is a graph with blocks \({B_1,B_2,…,B_n}\) such that for every \(i\), \(B_i\) and \(B_{i+1}\) have a common vertex, in such a way that the block-cut-vertex graph is a path. A chain graph having \({n}\) blocks \({B_1,B_2,…,B_n}\) is denoted by \([B_{1},B_{2},…,B_{n}]\). Let \({c_i}\) be the common vertex of \({B_i}\) and \({B_{i+1}}\), in \([{B_1,B_2,…,B_n}]\), \(1\le i \le n-1\). Consider a vertex of \({B_1}\), \({c_0 \ne c_1}\), and a vertex of \({B_n}\), \({c_n \ne c_{n-1}}\). Assume that \(B\) is another block graph and \(x\) and \(y\) are two different vertices in \(B\). The graph which is constructed by identifying \(c_0\) with \(x\) and \(c_n\) with \(y\) is called closed chain graph, and is denoted by \([{B_1,B_2,…,B_n,B}]_c\). By using pattern recognition and axiomatic deductive method we get results that some chain graphs with blocks of complete bipartite graphs are \(\alpha\)-labeling.

A cancellable fraction is a displayed numerator–denominator pair in which deleting the same decimal digit from the numerator and denominator leaves the represented rational number unchanged. We study vertical cancellations, where the deleted digits occur in the same decimal position in the numerator and denominator. Fix a nonzero digit \(h\). Let \(C_h(n)\) denote the number of marked vertical cancellations of the digit \(h\) among numerator–denominator pairs \((N,D)\), where \(N\) and \(D\) are both \(n\)-digit positive integers and \(D<N\). Here marked means that the cancelled position is part of the data. We prove that \[C_h(n)=O_h(n^2 10^{n-1}).\] Since the number of such displayed pairs \((N,D)\) is of order \(10^{2n}\), the proportion of vertically \(h\)-cancellable pairs is \(O_h(n^2/10^n)\). Thus vertical \(h\)-cancellability is exponentially rare. We also give an explicit family showing that \(C_h(n)\ge c_h10^n\) for all sufficiently large \(n\), and we include exact enumerations for small values of \(n\). Finally, we formulate a uniqueness conjecture asserting that almost every marked vertical cancellation belongs to a pair that is cancellable in exactly one vertical way.