Regarding Equitable Colorability Defect of Hypergraphs

Abstract

\noindent Azarpendar and Jafari in 2020 proved the following inequality $$\chi \left({\mathrm{KG}}^r(\mathcal{F},s)\right)\ge \lceil \frac{{\mathrm{ecd}}^r\left(\mathcal{F},\lfloor \frac{s}{2}\rfloor \right)}{r-1}\rceil ,$$ and noted that it is plausible that the above inequality remains true if one replaces $\lfloor \frac{s}{2}\rfloor$ with $s$. \noindent In this paper, considering the relation ${\mathrm{ecd}}^r\left(\mathcal{F},x\right)\ge {\mathrm{cd}}^r\left(\mathcal{F},x\right)$ which always holds, we show that even in the weaker inequality $$\chi \left({\mathrm{KG}}^r(\mathcal{F},s)\right)\ge \lceil \frac{{\mathrm{cd}}^r\left(\mathcal{F},\lfloor \frac{s}{2}\rfloor \right)}{r-1}\rceil ,$$ no number $x$ greater than $\lfloor \frac{s}{2}\rfloor$ could be replaced by $\lfloor \frac{s}{2}\rfloor$.