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.