The forb-flex method for odd coloring and proper conflict-free coloring of planar graphs

Abstract

We introduce a new technique useful for greedy coloring, which we call the forb-flex method, and apply it to odd coloring and proper conflict-free coloring of planar graphs. The odd chromatic number, denoted ${\chi }_o\left(G\right)$, is the smallest number of colors needed to properly color G such that every non-isolated vertex of G has a color appearing an odd number of times in its neighborhood. The proper conflict-free chromatic number, denoted ${\chi }_{\mathrm{PCF}}\left(G\right)$, is the smallest number of colors needed to properly color G such that every non-isolated vertex of G has a color appearing uniquely in its neighborhood. Our new technique works by carefully counting the structures in the neighborhood of a vertex and determining if a neighbor of a vertex can be recolored at the end of a greedy coloring process to avoid conflicts. Combining this with the discharging method allows us to prove ${\chi }_{\mathrm{PCF}}\left(G\right)\le 4$ for planar graphs of girth at least 11, and ${\chi }_o\left(G\right)\le 4$ for planar graphs of girth at least 10. These results improve upon the recent works of Cho, Choi, Kwon, and Park.