On odd colorings of sparse graphs
Tao Wang, Xiaojing Yang
TL;DR
This work advances the theory of odd colorings in sparse graphs by providing a complete extremal characterization at the boundary $\mathrm{mad}(G)=\frac{4c}{c+2}$ for $c\ge5$ in the context of PCF colorings, and by detailing the precise obstruction classes $\mathcal{G}_c$ (and $\mathcal{H}$ when $c=4$) that prevent such colorings. It develops a robust semi- PCF/ semi-odd framework and associated discharging techniques to connect maximum average degree bounds with colorability, yielding complete results for odd $4$-colorability and new planar graph guarantees for odd $6$-colorability under structural restrictions. The findings refine and extend Cranston's conjecture on mad, illuminate extremal configurations at equality, and yield practical implications for coloring sparse graphs and planar graphs under specific cycle/composition constraints. Overall, the paper significantly narrows the gap between conjectural bounds and exact extremal characterizations in odd and conflict-free 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.