On Schmerl's effective version of Brooks' theorem
نویسندگان
چکیده
منابع مشابه
A dual version of the Brooks group coloring theorem
Let G be a 2-edge-connected undirected graph, A be an (additive) Abelian group, and A = A − {0}. A graph G is A-connected if G has an orientation D(G) such that for every mapping b: V (G) → A satisfying v∈V (G) b(v) = 0, there is a function f : E(G) → A ∗ such that for each vertex v ∈ V (G), the sum of f over the edges directed out from v minus the sum of f over the edges directed into v eq...
متن کاملAn Improvement on Brooks’ Theorem
We prove that χ(G) ≤ max { ω(G),∆2(G), 5 6 (∆(G) + 1) } for every graph G with ∆(G) ≥ 3. Here ∆2 is the parameter introduced by Stacho that gives the largest degree that a vertex v can have subject to the condition that v is adjacent to a vertex whose degree is at least as large as its own. This upper bound generalizes both Brooks’ Theorem and the Ore-degree version of Brooks’ Theorem.
متن کاملBrooks' theorem on powers of graphs
We prove that for k ≥ 3, the bound given by Brooks’ theorem on the chromatic number of k-th powers of graphs of maximum degree ∆ ≥ 3 can be lowered by 1, even in the case of online list coloring.
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 کاملOn Brooks' Theorem for Sparse Graphs
Let G be a graph with maximum degree ∆(G). In this paper we prove that if the girth g(G) of G is greater than 4 then its chromatic number, χ(G), satisfies χ(G) ≤ (1 + o(1)) ∆(G) log ∆(G) where o(1) goes to zero as ∆(G) goes to infinity. (Our logarithms are base e.) More generally, we prove the same bound for the list-chromatic (or choice) number: χ l (G) ≤ (1 + o(1)) ∆(G) log ∆(G) provided g(G)...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Journal of Combinatorial Theory, Series B
سال: 1984
ISSN: 0095-8956
DOI: 10.1016/0095-8956(84)90041-8