Strong odd coloring in minor-closed classes

TL;DR

The paper investigates the strong odd chromatic number, $\operatorname{\chi_{so}}(G)$, of graphs and asks whether it is bounded within minor-closed graph classes. It situates $\operatorname{\chi_{so}}(G)$ among related colorings (including $\operatorname{\chi_{o}}(G)$ and $\chi(G^2)$) and presents foundational lower bounds via the $G_k$ construction, as well as concrete bounds for planar and outerplanar graphs (e.g., a planar example with $\operatorname{\chi_{so}}(G)=20$ and outerplanar bounds tightened to $8$). The main result demonstrates that for every proper minor-closed class $\mathcal{G}$ there exists a finite constant $c_{\mathcal{G}}$ with $\operatorname{\chi_{so}}(G) \le c_{\mathcal{G}}$ for all $G \in \mathcal{G}$, generalizing known boundedness beyond planar and outerplanar classes and outlining a three-step proof. This establishes a robust, class-dependent bound on strong odd colorings in a broad structural setting, with concrete implications for the colorability of complex graph families. The work also highlights near-complete determinations for outerplanar graphs and motivates further tightening for planar and related minor-closed classes.

Abstract

We show that the strong odd chromatic number on any proper minor-closed graph class is bounded by a constant. We almost determine the smallest such constant for outerplanar graphs.

Figures (1)

• Figure 1: (\ref{['fig:example_outerplanar']}) An outerplanar graph $G'$ with $\operatorname{\chi_{\rm so}}(G') = 7$. (\ref{['fig:example_planar']}) A planar graph $G"$ with $\operatorname{\chi_{\rm so}}(G") = 20$. (\ref{['fig:construction_odd_vs_strong_odd']}) The graph $G_3$ from \ref{['obs:nobinding']} with $\operatorname{\chi_{\rm so}}(G_3) = 2^3+1$. The black vertices in $G'$ and $G"$ have pairwise distinct colors in every strong odd coloring.

Theorems & Definitions (2)

• Proposition 3
• Theorem 4