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

Abstract

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 $χ_{\mathsf{o}}(G)$, 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 $χ_{\mathsf{PCF}}(G)$, 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 $χ_{\mathsf{PCF}}(G) \leq 4$ for planar graphs of girth at least 11, and $χ_{\mathsf{o}}(G) \leq 4$ for planar graphs of girth at least 10. These results improve upon the recent works of Cho, Choi, Kwon, and Park.

Figures (12)

• Figure 1: Illustration of the 9 neighbor types of $u$ together with $S[u]$ in red-dashed. Note that by \ref{['obs:odd']}\ref{['obs:odd - no odd vertex adjacent to a 2+ thread']}, the anchors of the $2^+$-threads are $4^+$-vertices, and the $t_{\mathsf{worst}}$-, $t_{\mathsf{bad}}$-, $t_{\mathsf{s\text{-}bad}}$-, $t_{\mathsf{good}}$-neighbors of $u$ are not anchors of $2^+$-threads.
• Figure 2:
• Figure 3: Different cases of the array representation $a_4 a_4 a_1 a_1$ as considered in Lemmas \ref{['obs:a_4a_4a_1a_1good3']} and \ref{['obs:a_4a_4a_1a_1semibad3']}. Note that in (II), $e$ is a $3^+$-vertex by \ref{['obs:odd']}\ref{['obs:odd - no odd vertex adjacent to a 2+ thread']}.
• Figure 4:
• Figure 5:

Theorems & Definitions (110)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Lemma 2.6
• proof
• Lemma 2.7