A Strengthening of Brooks' Theorem
نویسندگان
چکیده
منابع مشابه
A Strengthening of Brooks' Theorem for Line Graphs
We prove that if G is the line graph of a multigraph, then the chromatic number χ(G) of G is at most max {
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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...
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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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Let ∆(G) be the maximum degree of a graph G. Brooks’ theorem states that the only connected graphs with chromatic number χ(G) = ∆(G) + 1 are complete graphs and odd cycles. We prove a fractional analogue of Brooks’ theorem in this paper. Namely, we classify all connected graphs G such that the fractional chromatic number χf (G) is at least ∆(G). These graphs are complete graphs, odd cycles, C 2...
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Let G be a connected graph with maximum degree k (other than a complete graph or odd cycle), let W be a precolored set of vertices in G inducing a subgraph F , and let D be the minimum distance in G between components of F . If the components of F are complete graphs and D ≥ 8 (for k ≥ 4) or D ≥ 10 (for k = 3), then every proper k-coloring of F extends to a proper k-coloring of G. If the compon...
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ژورنال
عنوان ژورنال: Journal of Combinatorial Theory, Series B
سال: 1999
ISSN: 0095-8956
DOI: 10.1006/jctb.1998.1891