Coloring curves on surfaces

Abstract

We study the chromatic number of the curve graph of a surface. We show that the chromatic number grows like k log k for the graph of separating curves on a surface of Euler characteristic -k. We also show that the graph of curves that represent a fixed non-zero homology class is uniquely t-colorable, where t denotes its clique number. Together, these results lead to the best known bounds on the chromatic number of the curve graph. We also study variations for arc graphs and obtain exact results for surfaces of low complexity. Our investigation leads to connections with Kneser graphs, the Johnson homomorphism, and hyperbolic geometry.

Figures (6)

• Figure 1: (a) A right-angled hexagon $H$ and distinguished geodesics in $\mathbb D^2$. (b) Two ideal quadrilaterals and three reflections of $H$.
• Figure 2: An induced five cycle in $\mathcal{C}_v(S)$ when $g\ge 6$.
• Figure 3: The surface $S^2_g$. The Dehn twists about the red curves are Humphries generators for $\mathrm{PMod}(S^2_g)$. The blue curves comprise a maximal simplex in $\mathcal{C}_v(S^2_g)$, where $v$ denotes the peripheral homology class.
• Figure 4: The bounding pair map $\phi_{\alpha,\beta}$ applied to $\delta$, and the 2-chain $C(\delta,\phi_{\alpha,\beta}\cdot \delta)$ whose Euler measure is $\mathrm{genus}(\Sigma_1)$.
• Figure 5: A $(g+1)$-clique in $\mathcal{C}_v(S)$ with $v$ having a separating representative and not homologous to a boundary component.

Theorems & Definitions (98)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 2.1
• Lemma 2.2
• proof