For a path \( P_n \) of order \( n \) and for any odd integer \( k \), \( 1 \leq k \leq n – 3 \), Chartrand et al. have given an upper bound for the radio \( k \)-chromatic number of \( P_n \) as \( \frac{k^2+2k+1}{2} \). Here we improve this bound for \( \frac{n-4}{2} \leq k < \frac{2n-5}{3} \) and \( \frac{2n-5}{3} \leq k \leq n-3 \). They are \( \frac{k^2+k+4}{2} \) and \( \frac{k^2+k+2}{2} \), respectively. Also, we improve the lower bound of Kchikech et al. from \( \frac{k^2+3}{2} \) to \( \frac{k^2+5}{2} \) for odd integer \( k \), \( 3 \leq k \leq n-3 \).

In this paper, we obtain a necessary condition for the Skolem gracefulness of the disjoint union of \( k \) signed stars \( K_{1,r_i}, 1 \leq i \leq k \), which we call a \( k \)-signed star \( St(r_1,r_2,\ldots,r_k) \). We also present results on the Skolem gracefulness of the 2-signed star \( St(r_1,r_2) \).

In this paper, a definition of a variation of the standard notion of the line signed graph of a given signed graph is recalled from [14] and some fundamental results linking it to the notions of jump signed graphs [6] and adjacency signed graphs [21], especially with regard to their states of balance, consistency, and compatibility are obtained.

In this paper, we discuss how the addition of a new edge changes the irregularity strength in \( K_{m,m} \) and \( tC_4 \).

The goal of this article is to provide an overview of all the results currently known regarding the connectedness of path graphs. The proofs we present are only those that illustrate the different techniques employed in obtaining the results.

This is an expository paper addressed to readers with a small degree of familiarity with the field of graph theory and its techniques.

This paper presents some new results on permissible degree sets in polygon visibility graphs (PVGs). If the PVG has \( n \) vertices, we say it is an \( n \)-PVG. We also show some canonical construction techniques for PVGs with given degree sets.

In this paper, we present several graph theory related software systems that we have developed. These systems have been used in learning and research. The systems feature drawing and manipulation of graphs as well as execution of graph algorithms. The systems are: JGraph, a Java-based system for creating graphs and running graph algorithms; Colossus, a visibility graph system; Manohar, a system for computing graceful labelings of graphs (with special emphasis on trees); Graph Algorithm Constructor, which allows the creation of graph algorithms by drawing flow diagrams instead of writing source code. We also describe some examples in which the empirical data generated from these systems have allowed us to discover fundamental properties of graphs.

A simple acyclic graphoidal cover of a graph \( G \) is a collection \( \psi \) of paths in \( G \) such that every path in \( \psi \) has at least two vertices, every vertex of \( G \) is an internal vertex of at most one path in \( \psi \), every edge of \( G \) is in exactly one path in \( \psi \), and any two paths in \( \psi \) have at most one vertex in common. The minimum cardinality of a simple acyclic graphoidal cover of \( G \) is called the simple acyclic graphoidal covering number of \( G \) and is denoted by \( \eta_{as}(G) \). A simple acyclic graphoidal cover \( \psi \) of \( G \) with \( |\psi| = \eta_{as}(G) \) is called a minimum simple acyclic graphoidal cover of \( G \). Two minimum simple acyclic graphoidal covers \( \psi_1 \) and \( \psi_2 \) of \( G \) are said to be isomorphic if there exists an automorphism \( \alpha \) of \( G \) such that \( \psi = \{\alpha(P) : P \in \psi_1\} \). In this paper, we characterize trees, unicyclic graphs, and wheels in which any two minimum simple acyclic graphoidal covers are isomorphic.

In this paper, we study the domination number, the global domination number, the cographic domination number, the global cographic domination number, and the independent domination number of all the graph products which are non-complete extended \( p \)-sums (NEPS) of two graphs.

A sum composite labeling of a \((p,q)\) graph \( G = (V,E) \) is an injective function \( f : V(G) \to \{1,2,\dots,2p\} \) such that the function \( f^+ : E(G) \to C \) is also injective, where \( C \) denotes the set of all composite numbers and \( f^+ \) is defined by \( f^+(uv) = f(u) + f(v) \) for all \( uv \in E(G) \). A graph \( G \) is sum composite if there exists a sum composite labeling for \( G \). We give some classes of sum composite graphs and some classes of graphs which are not sum composite. We prove that it is possible to embed any graph \( G \) with a given property \( P \) in a sum composite graph which preserves the property \( P \), where \( P \) is the property of being connected, eulerian, hamiltonian, or planar. We also discuss the NP-completeness of the problem of determining the chromatic number and the clique number of sum composite graphs.