Regarding Equitable Colorability Defect of Hypergraphs

Abstract

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