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.