A simple graph \( G = (V(G), E(G)) \) admits an \( H \)-covering if every edge in \( E(G) \) belongs to a subgraph of \( G \) that is isomorphic to \( H \). An \((a,d)\)-\( H \)-\({antimagic\; total \;labeling}\) of \( G \) is a bijective function \( \xi : V(G) \cup E(G) \to \{1,2,\dots,|V(G)| + |E(G)|\} \) such that for all subgraphs \( H’ \) isomorphic to \( H \), the \( H \)-weights \( w(H’) = \sum_{v \in V(H’)} \xi(v) + \sum_{e \in E(H’)} \xi(e) \) constitute an arithmetic progression \( a, a+d, a+2d, \dots, a+(t-1)d \), where \( a \) and \( d \) are positive integers and \( t \) is the number of subgraphs of \( G \) isomorphic to \( H \). Additionally, the labeling \( \xi \) is called a \({super}\) \((a, d)\)-\( H \)-\({antimagic\; total\; labeling}\) if \( \xi(V(G)) = \{1, 2, \dots, |V(G)|\} \).

In this paper, we introduce the notion of \((a, d)\)-\( H \)-\({antimagic\; total\; labeling}\) and study some basic properties of such labeling. We provide an example of a family of graphs obtaining the labelings, that is providing \((a, d)\)-cycle-antimagic labelings of fans.

Let \( j \geq 2 \) be a natural number. For graphs \( G \) and \( H \), the size multipartite Ramsey number \( m_j(G, H) \) is the smallest natural number \( t \) such that any \( 2 \)-coloring by red and blue on the edges of \( K_{j \times t} \) necessarily forces a red \( G \) or a blue \( H \) as a subgraph. Let \( P_n \) be a path on \( n \) vertices. In this note, we determine the exact value of the size multipartite Ramsey number \( m_j(P_4, P_n) \) for \( n \geq 2 \).

Let \(j \geq 2\) be a natural number. For graphs \(G\) and \(H\), the size multipartite Ramsey number \(m_j(G, H)\) is the smallest natural number \(t\) such that any \(2\)-coloring by red and blue on the edges of \(K_{j \times t}\) necessarily forces a red \(G\) or a blue \(H\) as subgraph. Let \(P_n\) be a path on \(n\) vertices. In this note, we determine the exact value of the size multipartite Ramsey number \(m_j(P_4, P_n)\) for \(n \geq 2\).

In this paper, we determine the Ramsey number for the disjoint union of graphs versus a graph \( H \), especially \( R\left(\bigcup_{i=1}^{k} l_i S_{n_i}(1,1), W_6\right) \), \( R\left(\bigcup_{i=1}^{k} l_i S_{n_i}(1,2), W_6\right) \), and \( R\left(\bigcup_{i=1}^k l_i S_{n_i}, W_4\right) \).

In this paper, we discuss an upper bound for exponents of loopless asymmetric two-colored digraphs. If \( D \) is an asymmetric primitive two-colored digraph on \( n \) vertices, we show that \( \text{exp}(D) \leq 3n^2 + 2n – 2 \). For an asymmetric two-colored digraph \( D \) which contains a primitive two-colored cycle of length \( s \leq n \), we show its exponent is at most \( \frac{s^2 – 1}{2} + (s + 1)(n – s) \). We characterize such two-colored digraphs whose exponents equal \( \frac{s^2 – 1}{2} + (s + 1)(n – s) \) and show that the largest exponent of an asymmetric two-colored digraph lies in the interval \( \left[\frac{n^2 – 1}{2}, 3n^2 + 2n – 2\right] \) when \( n \) is odd, or \( \left[\frac{n^2}{2}, 3n^2 + 2n – 2\right] \) otherwise.

Let \( G = (V(G), E(G)) \) be a simple graph and let \( f \) be a function from \( V(G) \) to a subset of positive integers. An \( f \)-\({coloring}\) of \( G \) is a generalized edge-coloring such that every vertex \( v \in V(G) \) has at most \( f(v) \) edges colored with the same color. The minimum number of colors needed to define an \( f \)-coloring of \( G \) is called the \( f \)-\({chromatic\; index}\) of \( G \), and denoted by \( \chi_f'(G) \). The \( f \)-chromatic index of \( G \) is equal to \( \Delta_f(G) \) or \( \Delta_f(G) + 1 \), where \( \Delta_f(G) = \max \left\{ \frac{d(v)}{f(v)} \mid v \in V(G) \right\} \). \( G \) is called in the \({class-1}\), denoted by \( C_f1 \), if \( \chi_f'(G) = \Delta_f(G) \); otherwise \( G \) is called in the \({class-2}\), denoted by \( C_f2 \). In this paper, we show that the corona product of a cycle with either the complement of a complete graph, or a path, or a cycle is in \( C_f1 \).

For a simple graph \( G \) with the vertex set \( V \) and the edge set \( E \), a labeling \( \lambda: V \cup E \to \{1, 2, 3, \ldots, k\} \) is called a vertex-irregular total \( k \)-labeling of \( G \) if for any two different vertices \( x \) and \( y \) in \( V \) we have \( wt(x) \neq wt(y) \), where \( wt(x) = \lambda(x) + \sum_{xy \in E} \lambda(xy) \).

The total vertex-irregular strength, denoted by \( tvs(G) \), is the smallest positive integer \( k \) for which \( G \) has a vertex-irregular total \( k \)-labeling. In this paper, we determine the total vertex-irregular strength of a disjoint union of \( t \) copies of a path, denoted by \( tP_n \). We prove that for any \( t \geq 2 \),

\[
tvs(tP_n) =
\begin{cases}
t & \text{for } n = 1, \\
t+1 & \text{for } 2 \leq n \leq 3, \\
\lceil \frac{nt+1}{3} \rceil & \text{for } n \geq 4.
\end{cases}
\]

Let \( G = (V, E) \) be a graph with order \(|G|\) and size \(|E|\). An \((a, d)\)-vertex-antimagic total labeling is a bijection \( \alpha \) from the set of all vertices and edges to the set of consecutive integers \(\{1, 2, \ldots, |V| + |E|\}\), such that the weights of the vertices form an arithmetic progression with the initial term \( a \) and the common difference \( d \). If \( \alpha(V(G)) = \{1, 2, \ldots, |V|\} \) then we call the labeling super \((a, d)\)-vertex antimagic total. In this paper, we show some basic properties of such labelings on a disjoint union of regular graphs and demonstrate how to construct such labelings for some classes of graphs, such as cycles, generalized Petersen graphs, and circulant graphs, for \( d = 1 \).

In this paper, we determine the partition dimension of a complete multipartite graph \( K_{n_1, n_2, \ldots, n_r} \), namely \( pd(K_{n_1, n_2, \ldots, n_r}) \). We show that \( pd(K_{n_1, n_2, \ldots, n_r}) = r + n – 1 \) if \( n_i = n \) for \( 1 \leq i \leq r \), and \( pd(K_{n_1, n_2, \ldots, n_r}) = r + n – 2 \) for \( n = n_1 \geq n_2 \geq \ldots \geq n_r \). We also show that the partition dimension of the caterpillar graph \( C_n^m \) is \( m \) for \( n \leq m \) and \( m + 1 \) for \( n > m \), and the partition dimension of the windmill graph \( W_{2}^{n} \) is \( k \), where \( k \) is the smallest integer such that \( \binom{k}{2} \geq m \).

Let \( G \) be a graph with vertex set \( V = V(G) \) and edge set \( E = E(G) \), and let \( n = |V(G)| \) and \( e = |E(G)| \). A \({vertex-magic\; total\; labeling}\) (VMTL) of a graph is defined as a one-to-one mapping taking the vertices and edges onto the set of integers \(\{1, 2, \ldots, n+e\}\), with the property that the sum of the label on a vertex and the labels on its incident edges is a constant independent of the choice of vertex. In this paper, we present the vertex-magic total labeling of the disjoint union of \( t \) generalized Petersen graphs \(\bigcup_{j=1}^t P(n_j, m_j)\), and the disjoint union of \( t \) special circulant graphs \(\bigcup_{j=1}^t C_n(1, m_j)\).