In this paper, we extend the idea of magic labeling to directed graphs. In particular, a magic labeling of a digraph is the directed analog of a vertex-magic total labeling. Some elementary results are obtained and some infinite families of magic digraph labelings are exhibited.

Let \( P_h \) be a path on \( h \) vertices. A simple graph \( G = (V, E) \) admits a \( P_h \)-covering if every edge in \( E \) belongs to a subgraph of \( G \) that is isomorphic to \( P_h \). \( G \) is called \( P_h \)-magic if there is a total labeling \( f: V \cup E \to \{1, 2, \dots, |V| + |E|\} \) such that for each subgraph \( H’ = (V’, E’) \) of \( G \) that is isomorphic to \( P_h \), \( \sum_{v \in V’} f(v) + \sum_{e \in E’} f(e) \) is constant. When \( f(V) = \{1, 2, \dots, |V|\} \), we say that \( G \) is \( P_h \)-supermagic.

In this paper, we study some \( P_h \)-supermagic trees. We give some sufficient or necessary conditions for a tree to be \( P_h \)-supermagic. We also consider the \( P_h \)-supermagicness of special types of trees, namely shrubs and banana trees.

A \((p, q)\)-graph \( G \) is said to be multiplicative if its vertices can be assigned distinct positive integers so that the values of the edges, obtained as the products of the numbers assigned to their end vertices, are all distinct. Such an assignment is called a multiplicative labeling of \( G \). A multiplicative labeling is said to be \((a, r)\)-geometric if the values of the edges can be arranged as a geometric progression \( a, ar, ar^2, \dots, ar^{q-1} \). In this paper, we prove that some well-known classes of graphs are geometric for certain values of \( a, r \) and also initiate a study on the structure of finite \((a, r)\)-geometric graphs.

Given an integer \( \lambda \geq 2 \), a graph \( G = (V, E) \) and a spanning subgraph \( H \) of \( G \) (the backbone of \( G \)), a \( \lambda \)-backbone coloring of \( (G, H) \) is a proper vertex coloring \( V \to \{1, 2, \dots\} \) of \( G \), in which the colors assigned to adjacent vertices in \( H \) differ by at least \( \lambda \). We study the computational complexity of the problem “Given a graph \( G \) with a backbone \( H \), and an integer \( \ell \), is there a \( \lambda \)-backbone coloring of \( (G, H) \) with at most \( \ell \) colors?” Of course, this general problem is NP-complete. In this paper, we consider this problem for collections of pairwise disjoint complete graphs with order \( n \). We show that the complexity jumps from polynomially solvable to NP-complete between \( \ell = (n – 1)\lambda \) and \( \ell = (n – 1)\lambda + 1 \).

For a simple graph \( G = (V(G), E(G)) \) with the vertex set \( V(G) \) and the edge set \( E(G) \), a labeling \( \lambda: V(G) \cup E(G) \to \{1, 2, \dots, k\} \) is called an edge-irregular total \( k \)-labeling of \( G \) if for any two different edges \( e = e_1e_2 \) and \( f = f_1f_2 \) in \( E(G) \) we have \( wt(e) \neq wt(f) \) where \( wt(e) = \lambda(e_1) + \lambda(e) + \lambda(e_2) \). The total edge-irregular strength, denoted by \( tes(G) \), is the smallest positive integer \( k \) for which \( G \) has an edge-irregular total \( k \)-labeling. In this paper, we determine the total edge-irregular strength of the corona product of paths with some graphs.

For two given graphs \( G \) and \( H \), the Ramsey number \( R(G, H) \) is the smallest positive integer \( N \) such that for every graph \( F \) of order \( N \) the following holds: either \( F \) contains \( G \) as a subgraph or the complement of \( F \) contains \( H \) as a subgraph. In this paper, we shall study the Ramsey number \( R(T_n, W_m) \) for a star-like tree \( T_n \) with \( n \) vertices and a wheel \( W_m \) with \( m + 1 \) vertices and \( m \) odd. We show that the Ramsey number \( R(S_n, W_m) = 3n – 2 \) for \( n \geq 2m – 4, m \geq 5 \) and \( m \) odd, where \( S_n \) denotes the star on \( n \) vertices. We conjecture that the Ramsey number is the same for general trees on \( n \) vertices, and support this conjecture by proving it for a number of star-like trees.

A simple graph \( G(V, E) \) is called \( A \)-magic if there is a labeling \( f: E \to A^* \), where \( A \) is an Abelian group and \( A^* = A – \{0\} \), such that the induced vertex labeling \( f^*: V \to A \), defined as \( f^*(v) = \sum_{u \in N(v)} f(uv) = k \), for every \( v \in V \), is a constant in \( A \). In this paper, we show constructions of new classes of \( A \)-magic graphs from known \( A \)-magic graphs using labeling matrices.

For an ordered set \( W = \{w_1, w_2, \dots, w_k\} \) of vertices and a vertex \( v \) in a connected graph \( G \), the representation of \( v \) with respect to \( W \) is the ordered \( k \)-tuple \( r(v|W) = (d(v, w_1), d(v, w_2), \dots, d(v, w_k)) \) where \( d(x,y) \) represents the distance between the vertices \( x \) and \( y \). The set \( W \) is called a resolving set for \( G \) if every two vertices of \( G \) have distinct representations. A resolving set containing a minimum number of vertices is called a basis for \( G \). The dimension of \( G \), denoted by \( \text{dim}(G) \), is the number of vertices in a basis of \( G \). In this paper, we determine the dimensions of some corona graphs \( G \odot K_1 \), \( G \odot \overline{K}_m \) for any graph \( G \) and \( m \geq 2 \), and a graph with pendant edges more general than corona graphs \( G \odot \overline{K}_m \).

Let \( G = (V, E) \) be a simple, finite, and undirected graph. A sum labeling is a one-to-one mapping \( L \) from a set of vertices of \( G \) to a finite set of positive integers \( S \) such that if \( u \) and \( v \) are vertices of \( G \), then \( uv \) is an edge in \( G \) if and only if there is a vertex \( w \) in \( G \) and \( L(w) = L(u) + L(v) \). A graph \( G \) that has a sum labeling is called a sum graph. The minimal isolated vertex that is needed to make \( G \) a sum labeling is called the sum number of \( G \), denoted as \( \sigma(G) \). The sum number of a sum graph \( G \) is always greater than or equal to \( \delta(G) \), the minimum degree of \( G \). An optimum sum graph is a sum graph that has \( \sigma(G) = \delta(G) \). In this paper, we discuss sum numbers of finite unions of some families of optimum sum graphs, such as cycles and friendship graphs.

Let \( G = (V, E) \) be a simple and undirected graph with \( v \) vertices and \( e \) edges. An \( (a, d) \)-\emph{edge-antimagic total labeling} is a bijection \( f \) from \( V(G) \cup E(G) \) to the set of consecutive integers \( \{1, 2, \dots, v + e\} \) such that the weights of the edges form an arithmetic progression with initial term \( a \) and common difference \( d \). A super \( (a, d) \)-\emph{edge antimagic total labeling} is an edge antimagic total labeling \( f \) such that \( f(V(G)) = \{1, \dots, v\} \). In this paper, we solve some problems on edge antimagic total labeling, such as on paths and unicyclic graphs.