A study on the curling number of graph classes

TL;DR

This paper explores the properties of the curling number in various graph classes, including graph powers and products, extending the concept from sequences to graph degree sequences.

Contribution

It introduces new results on the behavior of the curling number under different graph operations and classes, expanding understanding of this sequence-based graph invariant.

Findings

Analyzed curling number of graph powers

Studied curling number of graph products

Examined effects of graph operations on curling number

Abstract

Given a finite nonempty sequence $S$ of integers, write it as $XY^k$, consisting of a prefix $X$ (which may possibly be empty), followed by $k$ copies of a non-empty string $Y$. Then, the greatest such integer $k$ is called the curling number of $S$ and is denoted by $cn(S)$. The concept of curling number of sequences has already been extended to the degree sequences of graphs to define the curling number of a graph. In this paper we study the curling number of graph powers, graph products and certain other graph operations.