English

Regarding Equitable Colorability Defect of Hypergraphs

Combinatorics 2022-12-05 v1

Abstract

\noindent Azarpendar and Jafari in 2020 proved the following inequality χ(KGr(F,s))ecdr(F,s2)r1,\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 s2\left\lfloor \frac{s}{2} \right\rfloor with ss. \noindent In this paper, considering the relation ecdr(F,x)cdr(F,x){\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 χ(KGr(F,s))cdr(F,s2)r1,\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 xx greater than s2\left\lfloor \frac{s}{2} \right\rfloor could be replaced by s2\left\lfloor \frac{s}{2} \right\rfloor.

Cite

@article{arxiv.2212.01358,
  title  = {Regarding Equitable Colorability Defect of Hypergraphs},
  author = {Saeed Shaebani},
  journal= {arXiv preprint arXiv:2212.01358},
  year   = {2022}
}
R2 v1 2026-06-28T07:20:46.640Z