On triangle-free list assignments

Abstract

We show that Bernshteyn's proof of the breakthrough result of Molloy that triangle-free graphs are choosable from lists of size (1+o(1))Δ/log⁡Δ can be adapted to yield a stronger result. In particular, one may prove that such list sizes are sufficient to colour any graph of maximum degree Δ provided that vertices sharing a common colour in their lists do not induce a triangle in G, which encompasses all cases covered by Molloy's theorem. This was thus far known to be true for lists of size (1000+o(1))Δ/log⁡Δ, as implied by a more general result due to Amini and Reed. We also prove that lists of length 2(r−2)Δlog2⁡log2⁡Δ/log2⁡Δ are sufficient if one replaces the triangle by any Kr with r≥4. All bounds presented are also valid within the more general setting of correspondence colourings.