Adaptable choosability of planar graphs with sparse short cycles

Given a (possibly improper) edge colouring F of a graph G , a vertex colouring of G is adapted to F if no colour appears at the same time on an edge and on its two endpoints. A graph G is called a d a p t a b l y k - c h o o s a b l e (for some positive integer k ) if for any list assignment L to the vertices of G , with | L ( v ) | ≥ k for all v , and any edge colouring F of G , G admits a colouring c adapted to F where c ( v ) ∈ L ( v ) for all v . This paper proves that a planar graph G is adaptably 3-choosable if any two triangles in G have distance at least 2 and no triangle is adjacent to a 4-cycle.