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.

We investigate the critical set of edge-magic labeling on caterpillar graphs and its application on secret sharing schemes. We construct a distribution scheme based on supervisory secret sharing schemes, which use the notion of critical sets to distribute the shares and reconstruct the key.

For given graphs \( G \) and \( H \), the Ramsey number \( R(G, H) \) is the least natural number \( n \) such that for every graph \( F \) of order \( n \) the following condition holds: either \( F \) contains \( G \) or the complement of \( F \) contains \( H \). In this paper, we improve the Ramsey number of paths versus Jahangirs. We also determine the Ramsey number \( R(\cup G, H) \), where \( G \) is a path and \( H \) is a Jahangir graph.

A total vertex irregular labeling of a graph G with v vertices and e edges is an assignment of integer labels to both vertices and edges so that the weights calculated at vertices are distinct.

The total vertex irregularity strength of G, denoted by tvs(G), is the minimum value of the largest label over all such irregular assignments.

In this paper, we consider the total vertex irregular labelings of wheels W_n, fans F_n, suns S_n and friendship graphs f_n.

tvs(W_n) = \lceil \frac{n+3}{4} \rceil \text{ for } n \geq 3,

tvs(F_n) = \lceil \frac{n+2}{4} \rceil \text{ for } n \geq 3,

tvs(S_n) = \lceil \frac{n+1}{2} \rceil \text{ for } n \geq 3,

tvs(f_n) = \lceil \frac{2n+2}{3} \rceil \text{ for all } n.

A graph is called supermagic if it admits a labeling of its edges by consecutive integers such that the sum of the labels of the edges incident with a vertex is independent of the particular vertex. In this paper we prove that the necessary conditions for an \(r\)-regular supermagic graph of order \(n\) to exist are also sufficient. All proofs are constructive and they are based on finding supermagic labelings of circulant graphs.A parameterized system consists of several similar processes whose number is determined by an input parameter. A challenging problem is to provide methods for the uniform verification of such systems, i.e., to show by a single proof that a system is correct for any value of the parameter.

This paper presents a method for verifying parameterized systems using predicate diagrams. Basically, predicate diagrams are graphs whose vertices are labelled with first-order formulas, representing sets of system states, and whose edges represent possible system transitions. These diagrams are used to represent the abstractions of parameterized systems described by specifications written in temporal logic.

This presented method integrates deductive verification and algorithmic techniques. Non-temporal proof obligations establish the correspondence between the original specification and the diagram, whereas model checking is used to verify properties over finite-state abstractions.

For any given graphs \( G \) and \( H \), we write \( F \rightarrow (G, H) \) to mean that any red-blue coloring of the edges of \( F \) contains a red copy of \( G \) or a blue copy of \( H \). A graph \( F \) is \((G, H)\)-minimal (Ramsey-minimal) if \( F \rightarrow (G, H) \) but \( F^* \not\rightarrow (G, H) \) for any proper subgraph \( F^* \subset F \). The class of all \((G, H)\)-minimal graphs is denoted by \( \mathcal{R}(G, H) \). In this paper, we will determine the graphs in \( \mathcal{R}(K_{1,2}, C_4) \).

Let \( G \) be a connected graph. For a vertex \( v \in V(G) \) and an ordered \( k \)-partition \( \Pi = (S_1, S_2, \dots, S_k) \) of \( V(G) \), the representation of \( v \) with respect to \( \Pi \) is the \( k \)-vector \( r(v|\Pi) = (d(v, S_1), d(v, S_2), \dots, d(v, S_k) ) \) where \( d(v, S_i) = \min_{w \in S_i} d(x, w) \) (\( 1 \leq i \leq k \)). The \( k \)-partition \( \Pi \) is said to be resolving if the \( k \)-vectors \( r(v|\Pi) \), \( v \in V(G) \), are distinct. The minimum \( k \) for which there is a resolving \( k \)-partition of \( V(G) \) is called the partition dimension of \( G \), denoted by \( pd(G) \). A resolving \( k \)-partition \( \Pi = \{ S_1, S_2, \dots, S_k \} \) of \( V(G) \) is said to be connected if each subgraph \( \langle S_i \rangle \) induced by \( S_i \) (\( 1 \leq i \leq k \)) is connected in \( G \). The minimum \( k \) for which there is a connected resolving \( k \)-partition of \( V(G) \) is called the connected partition dimension of \( G \), denoted by \( cpd(G) \). In this paper, the connected partition dimension of the unicyclic graphs is calculated and bounds are proposed.

Let \( G = (V, E) \) be a finite graph, where \( V(G) \) and \( E(G) \) are the (non-empty) sets of vertices and edges of \( G \). An \((a, d)\)-\emph{edge-antimagic total labeling} is a bijection \( \beta \) from \( V(G) \cup E(G) \) to the set of consecutive integers \( \{1, 2, \dots, |V(G)| + |E(G)|\} \) with the property that the set of all the edge-weights, \( w(uv) = \beta(u) + \beta(uv) + \beta(v) \), for \( uv \in E(G) \), is \( \{a, a + d, a + 2d, \dots, a + (|E(G)| – 1)d\} \), for two fixed integers \( a > 0 \) and \( d \geq 0 \). Such a labeling is super if the smallest possible labels appear on the vertices. In this paper, we investigate the existence of super \((a, d)\)-edge-antimagic total labelings for disjoint unions of multiple copies of a regular caterpillar.

The term mode graph was introduced by Boland, Kaufman, and Panrong to define a connected graph \( G \) such that, for every pair of vertices \( v, w \) in \( G \), the number of vertices with eccentricity \( e(v) \) is equal to the number of vertices with eccentricity \( e(w) \). As a natural extension to this work, the concept of an antimode graph was introduced to describe a graph for which, if \( e(v) \neq e(w) \), then the number of vertices with eccentricity \( e(v) \) is not equal to the number of vertices with eccentricity \( e(w) \). In this paper, we determine the existence of some classes of antimode graphs, namely equisequential and \((a, d)\)-antimode graphs.

By an \((a, d)\)-edge-antimagic total labeling of a graph \( G(V, E) \), we mean a bijective function \( f \) from \( V(G) \cup E(G) \) onto the set \( \{1, 2, \dots, |V(G)| + |E(G)|\} \) such that the set of all the edge-weights, \( w(uv) = f(u) + f(uv) + f(v) \), for \( uv \in E(G) \), is \( \{a, a+d, a+2d, \dots, a + (|E(G)| – 1)d\} \), for two integers \( a > 0 \) and \( d \geq 0 \).

In this paper, we study the edge-antimagic properties for the disjoint union of complete \( s \)-partite graphs.

We study the number of super edge-magic (bipartite) graphs from an asymptotic point of view.

It is well known that apart from the Petersen graph, there are no Moore graphs of degree 3. As a cubic graph must have an even number of vertices, there are no graphs of maximum degree 3 and \(\delta\) vertices less than the Moore bound, where \(\delta\) is odd. Additionally, it is known that there exist only three graphs of maximum degree 3 and 2 vertices less than the Moore bound. In this paper, we consider graphs of maximum degree 3, diameter \( D \geq 2 \), and 4 vertices less than the Moore bound, denoted as \((3, D, 4)\)-graphs. We obtain all non-isomorphic \((3, D, 4)\)-graphs for \( D = 2 \). Furthermore, for any diameter \( D \), we consider the girth of \((3, D, 4)\)-graphs. By a counting argument, it is easy to see that the girth is at least \( 2D – 2 \). The main contribution of this paper is that we prove that the girth of a \((3, D, 4)\)-graph is at least \( 2D – 1 \). Finally, for \( D > 4 \), we conjecture that the girth of a \((3, D, 4)\)-graph is \( 2D \).

A fast direct method for obtaining the incidence matrix of a finite projective plane of order \( n \) via \( n-1 \) mutually orthogonal \( n \times n \) Latin squares is described. Conversely, \( n-1 \) mutually orthogonal \( n \times n \) Latin squares are directly exhibited from the incidence matrix of a projective plane of order \( n \). A projective plane of order \( n \) can also be described via a digraph complete set of Latin squares, and a new procedure for doing this will also be described.