For positive integers \(k \leq n\), the crown \(C_{n,k}\) is the graph with vertex set \(\{a_0, a_1, \ldots, a_{n-1}, b_0, b_1, \ldots, b_{n-1}\}\) and edge set \(\{a_ib_j : 0 \leq i \leq n-1, j = i+1, i+2, \ldots, i+k \pmod{n}\}\). A caterpillar is a tree of order at least three which contains a path such that each vertex not on the path is adjacent to a vertex on the path. Being a connected bipartite graph, a caterpillar is balanced if the two parts of the bipartition of its vertices have equal size; otherwise, it is unbalanced. In this paper, we obtain the necessary and sufficient condition for balanced-caterpillar factorization of crowns. The criterion for unbalanced-caterpillar factorization of crowns is open. We also obtain the necessary and sufficient condition for directed caterpillar factorization of symmetric crowns.

This paper determines that the connectivity of the Cartesian product \(G_1 \square G_2\) of two graphs \(G_1\) and \(G_2\) is equal to \(\min\{\kappa_1v_2 + \kappa_2v_1, \delta_1 + \delta_2 \}\), where \(v_i, \kappa_i\), and \(\delta_i\) are the order, connectivity, and minimum degree of \(G_i\), respectively, for \(i = 1, 2\). Additionally, some necessary and sufficient conditions are given for \(G_1 \square G_2\) to be maximally connected and super-connected.

This paper deals with two types of graph labelings namely, super \((a, d)\)-edge antimagic total labeling and \((a, d)\)-vertex antimagic total labeling. We provide super \((a, d)\)-edge antimagic total labeling for disjoint unions of Harary graphs and disjoint unions of cycles. We also provide \((a,d)\)-vertex antimagic total labeling for disjoint unions of Harary graphs, disjoint unions of cycles, sun graphs and disjoint unions of sun graphs,

The existence question for a \(3\)-\((16,7,5)\) design is open, In this paper, we examine possible automorphisms of this design. We consider a minimum subset of basic permutations consisting of cycles of prime length \(p\) and prove that if a \(3\)-\((16,7,5)\) design exists, then it is either rigid or admits basic automorphisms with cycles of length \(2\) or \(3\).

We define a product summation of ordered partition \(f_j(n,m,r) = \sum{c_1^r c_2^r \ldots c_j^rc_{j+1} \ldots c_m}\), where the sum is over all positive integers \(c_1, c_2, \ldots, c_m\) with \(c_1 + c_2 + \cdots + c_m = n\) and \(0 \leq j \leq m\). We concentrate on \(f_m(n,m,r)\) in this paper. The main results are as follows:

(1) The generating function for \(f_m(n,m,r)\) and the explicit formula for \(f_m(n,m,2) , f_m(n,m,3)\) and \(f_m(n,m, 4)\) are obtained.

(2) The relationship between \(f_j(n,m,r)\) for \(r = 2,3\) and the Fibonacci and Lucas numbers is found.

It is shown that for \(2 \leq t \leq n-3\), a strict \(t\)-SB\((n,n-1)\) design does not exist, but for \(n \geq 3\), a non-strict \(2\)-SB\((n,n-1)\) design exists. The concept of large sets for Steiner triple systems is extended to SB designs and examples of large sets for SB designs are given.

It is shown that every well-defined solution to the second-order difference equation in the title, when \((A_n)_{n \in 0}\) is a two-periodic sequence such that \(\max\{A_0, A_1\} \geq 0\), is eventually periodic with period two. In the case \(\max\{A_0, A_1\} \leq 0\), it is shown the existence of unbounded solutions, by describing all solutions in terms of \(A_0\), \(A_1\), \(x_{-1}\), and \(x_0\).

This paper considers the folded hypercube \(FQ_n\) as an enhancement on the hypercube, and obtains some algebraic properties of \(FQ_n\). Using these properties, the authors show that for any two vertices \(x\) and \(y\) in \(FQ_n\), with distance \(d\) and any integers \(h \in \{d, n+1- d\}\) and \(l\) with \(h \leq l \leq 2^n – 1\), \(FQ_n\) contains an \(xy\)-path of length \(l\) and no \(xy\)-path of other length, provided that \(l\) and \(h\) have the same parity.

Let \(G\) be a \(2\)-tough graph on at least five vertices and let \(e_1, e_2\) be a pair of arbitrarily given edges of \(G\). Then
(a) There exists a \(2\)-factor in G containing \(e_1, e_2\).
(b) There exists a \(2\)-factor in G avoiding \(e_1, e_2\).
(c) There exists a \(2\)-factor in G containing \(e_1\) and avoiding \(e_2\).

In this paper, a sufficient condition is obtained for the global asymptotic stability of the following system of difference equations
\[x_{n+1} = \frac{x_ny_{n-1}}{x_ny_{n-1}+1} ,y_{n+1}=\frac{y_n x_{n-1}}{y_nx_{n-1} + 1} , \quad n = 0, 1, 2, \ldots,\]
where the initial values \((x_k, y_k) \in (0, \infty) (\text{for} k=-1,0)\).