A family of sets is called \(K\)-union distinct if all unions involving \(K\) or fewer members thereof are distinct. If a family of
sets is \(K\)-cover-free, then it is \(K\)-union distinct. In this paper, we recognize that this is only a sufficient condition and,
from this perspective, consider partially cover-free families of sets with a view to constructing union distinct families. The
role of orthogonal arrays and related combinatorial structures is explored in this context. The results are applied to find
efficient anti-collusion digital fingerprinting codes.

Let \(G\) be a \(2\)-edge-connected simple graph on \(n\) vertices, \(n \geq 3\). It is known that if \(G\) satisfies \(d(x) \geq \frac{n}{2}\) for every vertex \(x \in V(G)\), then \(G\) has a nowhere-zero \(3\)-flow, with several exceptions.In this paper, we prove that, with ten exceptions, all graphs with at most two vertices of degree less than \(\frac{n}{2}\) have nowhere-zero \(3\)-flows. More precisely, if \(G\) is a \(2\)-edge-connected graph on \(n\) vertices, \(n \geq 3\), in which at most two vertices have degree less than \(\frac{n}{2}\), then \(G\)
has a nowhere-zero \(3\)-flow if and only if \(G\) is not one of ten completely described graphs.

In this paper, we introduce the notion of right derivation of a weak BCC-algebra and investigate its related properties.
Additionally, we explore regular right derivations and d-invariants on weak BCC-ideals in weak BCC-algebras.

We investigate the Jacobsthal numbers \(\{J_n\}\) and Jacobsthal-Lucas numbers \(\{j_n\}\). Let \(\mathcal{J}_n = J_n \times j_n\) and \(\mathcal{J}_n = J_n + j_n\).In this paper, we give some determinantal and permanental representations for \(\mathcal{J}_n\) and \(\mathcal{J}_n\). Also, complex factorization formulas for the numbers are presented.

Let \(d\) be a fixed integer, \(0 \leq d \leq 2\), and let \(\mathcal{K}\) be a family of sets in the plane having simply connected union. Assume that for every countable subfamily \(\{K_n : n \geq 1\}\) of \(\mathcal{K}\), the union \(\cup\{K_n \geq 1\}\) is
starshaped via staircase paths and its staircase kernel contains a convex set of dimension at least \(d\). Then, \(\cup\{K:K \in \mathcal{K}\}\) has these properties as well.
In the finite case ,define function \(g\) on \((0, 1, 2) \) by \(g(0) = 2\), \(g(1) = g(2) = 4\). Let \(\mathcal{K}\) be a finite family of nonempty compact sets in the plane such that \(\cup\{K \in \mathcal{K}\}\) has a connected complement. For fixed \(d \in \{0, 1, 2\}\), assume that for every \(g(d)\) members of \(\mathcal{K}\), the corresponding union is starshaped via staircase paths and its staircase kernel contains a convex set of dimension at least \(d\). Then, \(\cup\{K \in \mathcal{K}\}\) also has these properties,also.
Most of these results are dual versions of theorems that hold for intersections of sets starshaped via staircase paths.The exceotion is the finite case above when \(d = 2\) .Surprisingly ,although the result for \(d=2\) holds for unique of sets, no analogue for intersections of sets is possible.

Let \(G\) be a simple connected graph containing a perfect matching.
\(G\) is said to be BM-extendable (bipartite matching extendable)
if every matching \(M\) which is a perfect matching of an induced
bipartite subgraph of \(G\) extends to a perfect matching of \(G\).

The BM-extendable cubic graphs are known to be \(K_{4}\) and \(K_{3,3}\).
In this paper, we characterize the 4-regular BM-extendable graphs.
We show that the only 4-regular BM-extendable graphs are \(K_{4,4}\) and
\(T_{4n}\), \(n \geq 2\), where \(T_{4n}\) is the graph on \(4n\) vertices
\(u_{i}\), \(v_{i}\), \(x_{i}\), \(y_{i}\), \(1 \leq i \leq n\), such that
\(\{u_{i}, v_{i}, x_{i}, y_{i}\}\) is a clique and
\(x_{i}u_{i+1}\), \(y_{i}v_{i+1} \in E(T_{4n})\) (mod \(n\)).

A rainbow coloring of the edges of a graph is a coloring such
that no two edges of the graph have the same color. The
anti-Ramsey number \(f(G, H)\) is the maximum number of colors
such that there is an \(H\)-anti-Ramsey edge coloring of \(G\), that is,
there exists no rainbow copy of the subgraph \(H\) of \(G\) in some
coloring of the edges of the host graph \(G\) with \(f(G, H)\) colors.

In this note, we exactly determine \(f(Q_5, Q_2)\) and \(f(Q_5, Q_3)\),
where \(Q_n\) is the \(n\)-dimensional hypercube.

The harmonic index \(H(G)\) of a graph \(G\) is defined as the sum
of weights \(\frac{2}{d(u) + d(v)}\) of all edges \(uv\) of \(G\), where
\(d(u)\) denotes the degree of a vertex \(u\) in \(G\).

In this paper, we establish sharp lower and upper bounds for the
harmonic index of bicyclic graphs and characterize the
corresponding extremal graphs.

For a graph \(G\), its Hosoya index is defined as the total number
of matchings in it, including the empty set. As one of the oldest and
well-studied molecular topological descriptors, the Hosoya index has
been extensively explored.

Notably, existing literature has primarily focused on its extremal
properties. In this note, we bridge a significant gap by establishing
sharp lower bounds for the Hosoya index in terms of other topological
indices.

We present a unified extension of alternating subsets to \(k\)-combinations
of \(\{1, 2, \ldots, n\}\) containing a prescribed number of sequences
of elements of the same parity. This is achieved by shifting attention
from parity-alternating elements to pairs of adjacent elements of the
same parity.

Enumeration formulas for both linear and circular combinations are
obtained by direct combinatorial arguments. The results are applied
to the enumeration of bit strings.