Some New Results on the Curling Number of Graphs

TL;DR

This paper investigates the properties of the curling number and compound curling number in the context of graph degree sequences, focusing on specific graph products to extend understanding of these graph invariants.

Contribution

It introduces new results on the behavior of curling numbers and compound curling numbers for various graph products, expanding the theoretical framework of these concepts.

Findings

Derived formulas for curling numbers of certain graph products

Established bounds for compound curling numbers in specific graph classes

Extended the concept of curling numbers to graph degree sequences

Abstract

Let $S=S_1S_2S_3\ldots S_n$ be a finite string. Write $S$ in the form $XYY\ldots Y=XY^k$, consisting of a prefix $X$ (which may be empty), followed by $k$ copies of a non-empty string $Y$. Then, the greatest value of this integer $k$ is called the curling number of $S$ and is denoted by $cn(S)$. Let the degree sequence of the graph $G$ be written as a string of identity curling subsequences say, $X^{k_1}_1\circ X^{k_2}_2\circ X^{k_3}_3 \ldots \circ X^{k_l}_l$. The compound curling number of $G$, denoted $cn^c(G)$ is defined to be, $cn^n(G) = \prod\limits^{l}_{i=1}k_i$. In this paper, we discuss the curling number and compound curling number of certain products of graphs.