A short proof of Brooks’ Theorem for vertex arboricity
نویسندگان
چکیده
منابع مشابه
Algebraic proof of Brooks’ theorem
We give a proof of Brooks’ theorem as well as its list coloring extension using the algebraic method of Alon and Tarsi.
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Brooks’ Theorem [R. L. Brooks, On Colouring the Nodes of a Network, Proc. Cambridge Philos. Soc. 37:194-197, 1941] states that every graph G with maximum degree ∆, has a vertex-colouring with ∆ colours, unless G is a complete graph or an odd cycle, in which case ∆ + 1 colours are required. Lovász [L. Lovász, Three short proofs in graph theory, J. Combin. Theory Ser. 19:269-271, 1975] gives an a...
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An equitable (t, k, d)-tree-coloring of a graph G is a coloring to vertices of G such that the sizes of any two color classes differ by at most one and the subgraph induced by each color class is a forest of maximum degree at most k and diameter at most d. The minimum t such that G has an equitable (t′, k, d)-tree-coloring for every t′ ≥ t is called the strong equitable (k, d)-vertex-arboricity...
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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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ژورنال
عنوان ژورنال: AKCE International Journal of Graphs and Combinatorics
سال: 2020
ISSN: 0972-8600,2543-3474
DOI: 10.1016/j.akcej.2019.03.005