On the topological lower bound for the multichromatic number

In 1976, Stahl [14] defined the m -tuple coloring of a graph G and formulated a conjecture on the multichromatic number of Kneser graphs. For m = 1 this conjecture is Kneser’s conjecture, which was proved by Lovász in 1978 [10]. Here we show that Lovász’s topological lower bound given in this way cannot be used to prove Stahl’s conjecture. We obtain that the strongest index bound only gives the trivial m ⋅ ω ( G ) lower bound if m ≥ | V ( G ) | . On the other hand, the connectivity bound for Kneser graphs is constant if m is sufficiently large. These findings provide new examples of graphs showing that the gaps between the chromatic number, the index bound and the connectivity bound can be arbitrarily large.