Circular choosability via combinatorial Nullstellensatz

A p-list assignment L of a graph G assigns to each vertex v of G a set L(v) ⊆ {0, 1, . . . , p − 1} of permissible colors. We say G is L-(p, q)-colorable if G has a (p, q)-coloring h such that h(v) ∈ L(v) for each vertex v. The circular list chromatic number χc,l(G) of a graph G is the infimum of those real numbers t for which the following holds: For any p, q, for any p-list assignment L with |L(v)| ≥ tq, G is L-(p, q)-colorable. We prove that if G has an orientation D which has no odd directed cycles, and L is a p-list assignment of G such that for each vertex v, |L(v)| = d+D(v)(2q − 1) + 1, then G is L-(p, q)-colorable. This implies that if G is a bipartite graph, then χc,l(G) ≤ 2dmad(G)/2e, where mad(G) is the maximum average degree of a subgraph of G. We further prove that if G is a connected bipartite graph which is not a tree, then χc,l(G) ≤ mad(G).