Sum Labelling Graphs of Maximum Degree Two

TL;DR

This paper characterizes the sum number of graphs with maximum degree two, which are disjoint unions of paths and cycles, expanding understanding of sum labelling in these specific graph classes.

Contribution

It provides a complete characterization of the sum number for graphs of maximum degree two, a class formed by disjoint unions of paths and cycles.

Findings

Sum number of graphs of maximum degree two is fully characterized.

Disjoint unions of paths and cycles determine the sum number.

Results extend understanding of sum labelling for specific graph classes.

Abstract

The concept of sum labelling was introduced in 1990 by Harary. A graph is a sum graph if its vertices can be labelled by distinct positive integers in such a way that two vertices are connected by an edge if and only if the sum of their labels is the label of another vertex in the graph. It is easy to see that every sum graph has at least one isolated vertex, and every graph can be made a sum graph by adding at most $n^2$ isolated vertices to it. The minimum number of isolated vertices that need to be added to a graph to make it a sum graph is called the sum number of the graph. The sum number of several prominent graph classes (e.g., cycles, trees, complete graphs) is already well known. We examine the effect of taking the disjoint union of graphs on the sum number. In particular, we provide a complete characterization of the sum number of graphs of maximum degree two, since every such graph is the disjoint union of paths and cycles.