Brooks’s Theorem for Measurable Colorings

Throughout, by a graph we mean a simple undirected graph, where the degree of a vertex is its number of neighbors, and a d-coloring is a function assigning each vertex one of d colors so that adjacent vertices are mapped to different colors. This paper examines measurable analogues of Brooks’s Theorem. While a straightforward compactness argument extends Brooks’s Theorem to infinite graphs, such an argument cannot produce a coloring with desirable measurability properties such as being being μ-measurable with respect to some probability measure, or being Baire measurable with respect to some Polish topology. Indeed, in this setting a straightforward analogue of the d = 2 case of Brooks’s Theorem does not hold for either of these measurability notions. Let S : T → T be an irrational rotation of the unit circle T, and let GS be the graph on T rendering adjacent each point x ∈ T and its image S(x) under S. Then GS is acyclic, each vertex has degree 2, and an easy ergodicity argument shows that GS has no Lebesgue measurable 2-coloring: since S is measure preserving, the color sets would have to have equal measure, but since S2 is ergodic, the color sets would each have to be null or conull. Similarly, GS has no Baire measurable 2-coloring (see Section 8). Our main result is the following measurable analogue of Brooks’s theorem for the case d ≥ 3. Recall that a standard Borel space is a set X equipped with a σ-algebra generated by a Polish (separable, completely metrizable) topology. Then a Borel graph G is a graph whose vertices are the elements of some standard Borel space X, and whose edge relation is Borel as a subset of X ×X.