A graph is -colorable if can be partitioned into two sets and so that is an independent set and has maximum degree at most 1. The problem of verifying whether a graph is -colorable is NP-complete even in the class of planar graphs of girth 9.
The maximum average degree, , of a graph is the maximum of over all subgraphs of . It was proved recently that every graph with is -colorable, and this is sharp. This yields that every planar graph with girth at least 12 is -colorable.
Let denote the supremum of such that for some constant every graph with girth and for every is -colorable. By the above, . We find the exact value for and : . In fact, we also find the best possible values of and : every triangle-free graph with for every is -colorable, and there are infinitely many not -colorable graphs with girth 5, and for every proper subgraph of . A corollary of our result is that every planar graph of girth 11 is -colorable. This answers a half of a question by Dorbec, Kaiser, Montassier and Raspaud. In a companion paper, we show that for every , and resolve some similar problems for the so-called -colorings generalizing -colorings.