Proper conflict-free choosability of sparse graphs with girth at least seven

Source abstract

A proper conflict-free coloring is a proper vertex coloring in which every non-isolated vertex has a color appearing exactly once in its open neighborhood. We prove that every finite simple graph with girth at least 7 and maximum average degree less than 8/3 admits a proper conflict-free coloring from arbitrary vertex lists of size at least the vertex degree plus 2. Consequently, every planar graph of girth at least 8 is proper conflict-free (degree+2)-choosable, improving the sufficient girth bound of 9 obtained from the earlier 18/7 maximum-average-degree theorem. The proof uses local extension lemmas for short threads, including threads with a common boundary endpoint. Two-element control sets and an incidence count yield a weighted thread inequality, which supplies the required bound on the charge sent by each vertex in a discharging argument.