A Generalization of Kneser's Conjecture

We investigate some coloring properties of Kneser graphs. A star-free coloring is a proper coloring $c:V(G)\to \Bbb{N}$ such that no path with three vertices may be colored with just two consecutive numbers. The minimum positive integer $t$ for which there exists a star-free coloring $c: V(G) \to \{1,2,..., t\}$ is called the star-free chromatic number of $G$ and denoted by $\chi_s(G)$. In view of Tucker-Ky Fan's lemma, we show that for any Kneser graph ${\rm KG}(n,k)$ we have $\chi_s({\rm KG}(n,k))\geq \max\{2\chi({\rm KG}(n,k))-10, \chi({\rm KG}(n,k))\}$ where $n\geq 2k \geq 4$. Moreover, we show that $\chi_s({\rm KG}(n,k))=2\chi({\rm KG}(n,k))-2=2n-4k+2$ provided that $n \leq {8\over 3}k$. This gives a partial answer to a conjecture of [12]. Also, we conjecture that for any positive integers $n\geq 2k \geq 4$ we have $\chi_s({\rm KG}(n,k))= 2\chi({\rm KG}(n,k))-2$.