Exponentially many 4‐list‐colorings of triangle‐free graphs on surfaces

Thomassen proved that every planar graph G on n vertices has at least 2 n / 9distinct L ‐colorings if L is a 5‐list‐assignment for G and at least 2 n / 10000distinct L ‐colorings if L is a 3‐list‐assignment for G and G has girth at least five. Postle and Thomas proved that if G is a graph on n vertices embedded on a surface Σ of genus g , then there exist constants ε , c g > 0 such that if G has an L ‐coloring, then G has at leastc g 2 ε ndistinct L ‐colorings if L is a 5‐list‐assignment for G or if L is a 3‐list‐assignment for G and G has girth at least five. More generally, they proved that there exist constants ε , α > 0 such that if G is a graph on n vertices embedded in a surface Σ of fixed genus g , H is a proper subgraph of G , and ϕ is an L ‐coloring of H that extends to an L ‐coloring of G , then ϕ extends to at least 2 ε ( n − α ( g + | V ( H ) | ) )distinct L ‐colorings of G if L is a 5‐list‐assignment or if L is a 3‐list‐assignment and G has girth at least five. We prove the same result if G is triangle‐free and L is a 4‐list‐assignment of G , where ε = 1 8 , and α = 130 .