Odd coloring of sparse graphs and planar graphs

An odd c-coloring of a graph is a proper c-coloring such that each non-isolated vertex has a color appearing an odd number of times on its neighborhood. This concept was introduced very recently by Petruševski and Škrekovski and has attracted considerable attention. Cranston investigated odd colorings of graphs with bounded maximum average degree, and conjectured that every graph G with mad(G)≤4c−4c+1 has an odd c-coloring for c≥4, and proved the conjecture for c∈{5,6}. In particular, planar graphs with girth at least 7 and 6 have an odd 5-coloring and an odd 6-coloring, respectively. We completely resolve Cranston's conjecture. For c≥7, we show that the conjecture is true, in a stronger form that was implicitly suggested by Cranston, but for c=4, we construct counterexamples, which all contain 5-cycles. On the other hand, we show that a graph G with mad(G)