Optimal $\ddot{I}‡$-boundness of $\ell$-holed graphs

Abstract

A graph is {\em{$\ell$-holed}} if all of its induced cycles of length at least four have length exactly $\ell$. In the paper, we prove that if $G$ is an $\ell$-holed graph with odd $\ell \ge 7$, then $\ddot{I}‡(G)\le \lceil \frac{\ell }{\ell -1}\ddot{I}‰(G)\rceil$. This result is sharp.

arXiv admin note: text overlap with arXiv:2110.09970, arXiv:2410.19005 by other authors