On the Curling Number of the Mycielskian of Certain Graphs

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 \( \text{cn}(S) = \max\{k_i : 1 \leq i \leq r\} \). Analogous to this concept, the degree sequence of the graph \( G \) can be written as a string \( X_1^{k_1} \circ X_2^{k_2} \circ X_3^{k_3} \circ \dots \circ X_r^{k_r} \). 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.