Let \(G\) be a graph, and let \(a\) and \(b\) be integers with \(1 \leq a \leq b\). An \([a, b]\)-factor of \(G\) is defined as a spanning subgraph \(F\) of \(G\) such that \(a \leq d_F(v) \leq b\) for each \(v \in V(G)\). In this paper, we obtain a sufficient condition for a graph to have \([a, b]\)-factors including given edges, extending a well-known sufficient condition for the existence of a \(k\)-factor.

We introduce the domination polynomial of a graph \(G\). The domination polynomial of a graph \(G\) of order \(n\) is defined as \(D(G, x) = \sum_{i=\gamma(G)}^{n} d(G, i)x^i\), where \(d(G, i)\) is the number of dominating sets of \(G\) of size \(i\), and \(\gamma(G)\) is the domination number of \(G\). We obtain some properties of \(D(G, x)\) and its coefficients, and compute this polynomial for specific graphs.

For a tree \(T\), \(Leaf(T)\) denotes the set of leaves of \(T\), and \(T – Leaf(T)\) is called the stem of \(T\). For a graph \(G\) and a positive integer \(m\), \(\sigma_m(G)\) denotes the minimum degree sum of \(m\) independent vertices of \(G\). We prove the following theorem: Let \(G\) be a connected graph and \(k \geq 2\) be an integer. If \(\sigma_3(G) \geq |G| – 2k + 1\), then \(G\) has a spanning tree whose stem has at most \(k\) leaves.

A proper vertex coloring of a graph is equitable if the sizes of color classes differ by at most \(1\). The equitable chromatic threshold of a graph \(G\), denoted by \(\chi_m^*(G)\), is the minimum \(k\) such that \(G\) is equitably \(k’\)-colorable for all \(k’ > k\). Let \(G \times H\) denote the direct product of graphs \(G\) and \(H\). For \(n \geq m \geq 2\), we prove that \(\chi_m^*(K_m \times K_n)\) equals \(\left\lceil \frac{mn}{m+1} \right\rceil\) if \(n \equiv 2, \ldots, m \pmod{m+1}\), and equals \(m\left\lceil \frac{n}{s^*} \right\rceil\) if \(n \equiv 0, 1 \pmod{m+1}\), where \(s^*\) is the minimum positive integer such that \(s^* \nmid n\) and \(s^* \geq m+2\).

For an undirected graph \(G\) and a natural number \(n\), a \(G\)-design of order \(n\) is an edge partition of the complete graph \(K_n\) with \(n\) vertices into subgraphs \(G_1, G_2, \ldots\), each isomorphic to \(G\). A set \(T \subset V(K_n)\) is called a blocking set if it intersects the vertex set \(V(G_i)\) of each \(G_i\) in the decomposition but contains none of them. Extending previous work [J. Combin. Designs \(4 (1996), 135-142]\), where the authors proved that cycle designs admit no blocking sets, we establish that this result holds for all graphs \(G\). Furthermore, we show that for every graph \(G\) and every integer \(k \geq 2\), there exists a non-\(k\)-colorable \(G\)-design.

Let \(G\) be a planar graph with maximum degree \(\Delta(G)\). The least integer \(k\) such that \(G\) can be partitioned into \(k\) edge-disjoint forests, where each component is a path of length at most \(2\), is called the linear \(2\)-arboricity of \(G\), denoted by \(la_2(G)\). We establish new upper bounds for the linear \(2\)-arboricity of certain planar graphs.

A graph \(G\) of order \(n\) is called a bicyclic graph if \(G\) is connected and the number of edges of \(G\) is \(n+ 1\). In this paper, we study the lexicographic ordering of bicyclic graphs by spectral moments. For each of the three basic types of bicyclic graphs on a fixed number of vertices maximal and minimal graphs in the mentioned order are determined.

An edge irregular total \(k\)-labeling of a graph \(G = (V, E)\) is a labeling \(f: V \cup E \to \{1, 2, \ldots, k\}\) such that the total edge-weights \(wt(xy) = f(x) + f(xy) + f(y)\) are distinct for all pairs of distinct edges. The minimum \(k\) for which \(G\) has an edge irregular total \(k\)-labeling is called the total edge irregularity strength of \(G\). In this paper, we determine the exact value of the total edge irregularity strength of the Cartesian product of two paths \(P_n\) and \(P_m\). Our result provides further evidence supporting a recent conjecture of Ivančo and Jendrol.

For a vertex set \(S\) with cardinality at least \(2\) in a graph \(G\), a tree connecting \(S\), known as a Steiner tree or \(S\)-tree, is required. Two \(S\)-trees \(T\) and \(T’\) are internally disjoint if \(V(T) \cap V(T’) = S\) and \(E(T) \cap E(T’) = \emptyset\). Let \(\kappa_G(G)\) denote the maximum number of internally disjoint Steiner trees connecting \(S\) in \(G\). The generalized \(k\)-connectivity \(\kappa_k(G)\) of \(G\), introduced by Chartrand et al., is defined as \(\min_{S \subseteq V(G), |S|=k} \kappa_G(S)\). This paper establishes a sharp upper bound for generalized \(k\)-connectivity. Furthermore, graphs of order \(n\) with \(\kappa_3(G) = n-2,n-3\) are characterized.

A hypergraph \(\mathcal{H}\) is said to be \(p\)-Helly when every \(p\)-wise intersecting partial hypergraph \(\mathcal{H}’\) of \(H\) has nonempty total intersection. Such hypergraphs were characterized by Berge and Duchet in 1975, and since then they have appeared in various contexts, particularly for \(p=2\), where they are known as Helly hypergraphs. An interesting generalization due to Voloshin considers both the number of intersecting sets and their intersection sizes: a hypergraph \(\mathcal{H}\) is \((p,q,s)\)-Helly if every \(p\)-wise \(q\)-intersecting partial hypergraph \(\mathcal{H}’\) of \(H\) has total intersection of cardinality at least \(s\). This work proposes a characterization for \((p,q,s)\)-Helly hypergraphs, leading to an efficient algorithm for recognizing such hypergraphs when \(p\) and \(q\) are fixed parameters.