On odd colorings of sparse graphs

Abstract

An \emph{odd $c$-coloring} of a graph is a proper $c$-coloring such that each non-isolated vertex has a color appearing an odd number of times within its open neighborhood. A \emph{proper conflict-free $c$-coloring} of a graph is a proper $c$-coloring such that each non-isolated vertex has a color appearing exactly once within its neighborhood. Clearly, every proper conflict-free $c$-coloring is also an odd $c$-coloring. Cranston conjectured that every graph $G$ with maximum average degree $\text{mad}(G) < \frac{4c}{c+2}$ (where $c \geq 4$) has an odd $c$-coloring, and he proved this conjecture for $c \in \{5, 6\}$. Note that the bound $\frac{4c}{c+2}$ is best possible. Cho et al. solved Cranston's conjecture for $c \geq 5$, strengthening the result by transitioning from odd $c$-coloring to proper conflict-free $c$-coloring. However, they did not provide all the extremal non-colorable graphs $G$ with $\text{mad}(G) = \frac{4c}{c+2}$, which remains an open question of interest. In this paper, we tackle this intriguing extremal problem. We aim to characterize all non-proper conflict-free $c$-colorable graphs $G$ with $\text{mad}(G) = \frac{4c}{c+2}$. For the case of $c=4$, Cranston's conjecture is not true, as evidenced by the existence of a counterexample: a graph whose every block is a $5$-cycle. Cho et al.\ proved that a graph $G$ with $\text{mad}(G) < \frac{22}{9}$ and no induced $5$-cycles has an odd $4$-coloring. We improve this result by proving that a graph $G$ with $\text{mad}(G) \leq \frac{22}{9}$ (with equality allowed) is not odd $4$-colorable if and only if $G$ belongs to a specific class of graphs. On the other hand, Cho et al.\ established that a planar graph with girth at least $5$ has an odd $6$-coloring; we improve it by proving that a planar graph without $4^{-}$-cycles adjacent to $7^{-}$-cycles also has an odd $6$-coloring.