Shifts of the stable Kneser graphs and hom-idempotence

A graph G is said to be hom-idempotent if there is a homomorphism from G2 to G, and weakly hom-idempotent if for some n ≥ 1 there is a homomorphism from Gn+1 to Gn. Larose et al. (1998) proved that Kneser graphs KG(n, k) are not weakly hom-idempotent for n ≥ 2k + 1, k ≥ 2. For s ≥ 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 KG(n, k)s−stab and we show that 2-stable Kneser graphs are not weakly hom-idempotent, for n ≥ 2k + 2, k ≥ 2. Moreover, for s, k ≥ 2, we prove that sstable Kneser graphs KG(ks+1, k)s−stab are circulant graphs and so hom-idempotent graphs. Finally, for s ≥ 3, we show that s-stable Kneser graphs KG(2s+2, 2)s−stab are cores, notχ-critical, not homidempotent and their chromatic number is equal to s + 2. © 2016 Elsevier Ltd. All rights reserved.