2024/01/26 by J. M. Anderson, Herman Chau, Anderson, James +15 · 1 citation
Computer Science · #Advanced Graph Theory Research #Combinatorics (math.CO) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2401.14590
openalex publication_date 2024/01/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce a new tool 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 χo(G), 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 χPCF(G), 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 tool 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 χPCF(G) ≤ 4 for planar graphs of girth at least 11, and χo(G) ≤ 4 for planar graphs of girth at least 10. These results improve upon the recent works of Cho, Choi, Kwon, and Park.