Coloring, sparseness and girth

An r-augmented tree is a rooted tree plus r edges added from each leaf to ancestors. For d, g, r a N, we construct a bipartite r-augmented complete d-ary tree having girth at least g. The height of such trees must grow extremely rapidly in terms of the girth. Using the resulting graphs, we construct sparse non-k-choosable bipartite graphs, showing that maximum average degree at most 2(k - 1) is a sharp sufficient condition for k-choosability in bipartite graphs, even when requiring large girth. We also give a new simple construction of non-k-colorable graphs and hypergraphs with any girth.