Shifts of the Stable Kneser Graphs and Hom-Idempotence

A graph $G$ is said to be {\em hom-idempotent} if there is a homomorphism from $G^2$ to $G$, and {\em weakly hom-idempotent} if for some $n \geq 1$ there is a homomorphism from $G^{n+1}$ to $G^n$. Larose et al. [{\em Eur. J. Comb. 19:867-881, 1998}] proved that Kneser graphs $\operatorname{KG}(n,k)$ are not weakly hom-idempotent for $n \geq 2k+1$, $k\geq 2$. For $s \geq 2$, we characterize all the shifts (i.e., automorphisms of the graph that map every vertex to one of its neighbors) of $s$-stable Kneser graphs $\operatorname{KG}(n,k)_{s-\operatorname{stab}}$ and we show that $2$-stable Kneser graphs are not weakly hom-idempotent, for $n \geq 2k+2$, $k \geq 2$. Moreover, for $s,k\geq 2$, we prove that $s$-stable Kneser graphs $\operatorname{KG}(ks+1,k)_{s-\operatorname{stab}}$ are circulant graphs and so hom-idempotent graphs. Finally, for $s \geq 3$, we show that $s$-stable Kneser graphs $\operatorname{KG}(2s+2,2)_{s-\operatorname{stab}}$ are cores, not $\chi$-critical, not hom-idempotent and their chromatic number is equal to $s+2$.