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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 nonzero 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.

The orientation preserving topological symmetry group of a graph embedded in the 3-sphere is the subgroup of the automorphism group of the graph consisting of those automorphisms which can be induced by an orientation preserving homeomorphism of the ambient space. We characterize all possible orientation preserving topological symmetry groups of embeddings of complete graphs in the 3-sphere.

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COLORING CURVES ON SURFACES

TOPOLOGICAL SYMMETRY GROUPS OF COMPLETE GRAPHS IN THE 3-SPHERE

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2 results

COLORING CURVES ON SURFACES

TOPOLOGICAL SYMMETRY GROUPS OF COMPLETE GRAPHS IN THE 3-SPHERE