Coloring, sparseness, and girth · arXivDesk1412.8002Dec 26, 2014Slight adjustments, including simplifying the proofs of Lemmas 2.2 and 2.3 and the arguments for large girth
Coloring, sparseness, and girth
Noga Alon, Alexandr Kostochka, Benjamin Reiniger, Douglas B. West, Xuding Zhu
Abstract
An r-augmented tree is a rooted tree plus r edges added from each leaf to ancestors. For d,g,r∈N, we construct a bipartite r-augmented complete d
-ary tree having girth at least
. The height of such trees must grow extremely rapidly in terms of the girth. Using the resulting graphs, we construct sparse non-
-choosable bipartite graphs, showing that maximum average degree at most
is a sharp sufficient condition for
-choosability in bipartite graphs, even when requiring large girth. We also give a new simple construction of non-
-colorable graphs and hypergraphs with any girth
.