منابع مشابه
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, suc...
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If Γ is an infinite group with finite symmetric generating set S, we consider the graph G(Γ, S) on [0, 1]Γ by relating two distinct points if an element of s sends one to the other via the shift action. We show that, aside from the cases Γ = Z and Γ = (Z/2Z) ∗ (Z/2Z), G(Γ, S) satisfies a measure-theoretic version of Brooks’ theorem: there is a G(Γ, S)-invariant conull Borel set B ⊆ [0, 1]Γ and ...
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The measurable list chromatic number of a graph G is the smallest number ξ such that if each vertex v of G is assigned a set L(v) of measure ξ in a fixed atomless measure space, then there exist sets c(v) ⊆ L(v) such that each c(v) has measure one and c(v) ∩ c(v′) = ∅ for every pair of adjacent vertices v and v′. We provide a simpler proof of a measurable generalization of Hall’s theorem due to...
متن کاملBrooks' Theorem and Beyond
We collect some of our favorite proofs of Brooks’ Theorem, highlighting advantages and extensions of each. The proofs illustrate some of the major techniques in graph coloring, such as greedy coloring, Kempe chains, hitting sets, and the Kernel Lemma. We also discuss standard strengthenings of vertex coloring, such as list coloring, online list coloring, and Alon–Tarsi orientations, since analo...
متن کاملNo Quantum Brooks' Theorem
First, I introduce quantum graph theory. I also discuss a known lower bound on the independence numbers and derive from it an upper bound on the chromatic numbers of quantum graphs. Then, I construct a family of quantum graphs that can be described as tropical, cyclical, and commutative. I also define a step logarithm function and express with it the bounds on quantum graph invariants in closed...
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ژورنال
عنوان ژورنال: Forum of Mathematics, Sigma
سال: 2016
ISSN: 2050-5094
DOI: 10.1017/fms.2016.14