# Special cases of the orbifold version of Zvonkine's $r$-ELSV formula

## TL;DR

This paper proves specific instances of Zvonkine's r-ELSV formula in orbifold settings, focusing on cases with r=2 for all genera and any r for genus zero, advancing understanding of orbifold Hurwitz numbers.

## Contribution

It establishes the orbifold version of Zvonkine's r-ELSV formula in two particular cases, expanding the theoretical framework for orbifold Hurwitz numbers.

## Key findings

- Proved orbifold Zvonkine's r-ELSV formula for r=2 across all genera.
- Validated the formula for any r in genus zero.
- Enhanced the mathematical understanding of orbifold Hurwitz numbers.

## Abstract

We prove the orbifold version of Zvonkine's $r$-ELSV formula in two special cases: the case of $r=2$ (complete $3$-cycles) for any genus $g\geq 0$ and the case of any $r\geq 1$ for genus $g=0$.