Odd coloring of sparse graphs and planar graphs

An odd c-coloring of a graph is proper such that each non-isolated vertex has color appearing an number times on its neighborhood. This concept was introduced very recently by Petruševski and Škrekovski attracted considerable attention. Cranston investigated colorings graphs with bounded maximum average degree, conjectured every G mad(G)≤4c−4c+1 for c≥4, proved the conjecture c∈{5,6}. In particular, planar girth at least 7 6 have 5-coloring 6-coloring, respectively. We completely resolve Cranston's conjecture. For c≥7, we show true, in stronger form implicitly suggested Cranston, but c=4, construct counterexamples, which all contain 5-cycles. On other hand, mad(G)<229 no induced 5-cycles 4-coloring. implies 11 also prove 5 6-coloring.