On the Curling Number of the Mycielskian of Certain Graphs

Abstract

Let $S=S_1{S}_2{S}_3\dots S_n$ be a finite string which can be written in the form $X_1^{k_1}{X}_2^{k_2}\dots X_r^{k_r}$, where $X_i^{k_i}$ is the $k_i$ copies of a non-empty string $X_i$ and each $k_i$ is a non-negative integer. Then, the curling number of the string $S$, denoted by $\text{cn}(S)$, is defined to be $\text{cn}(S)=max\{k_i:1\le i\le r\}$. Analogous to this concept, the degree sequence of the graph $G$ can be written as a string $X_1^{k_1}\circ X_2^{k_2}\circ X_3^{k_3}\circ \cdots \circ X_r^{k_r}$. The compound curling number of $G$, denoted ${\text{cn}}^c(G)$, is defined to be ${\text{cn}}^c(G)=\prod _{i=1}^r{k}_i.$ In this paper, the curling number and compound curling number of the powers of the Mycielskian of certain graphs are discussed.