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The Effective Version of Brooks' Theorem [PDF]
Canadian Journal of Mathematics, 1982 One of the fundamental results on graph coloring is the following classical theorem of Brooks.BROOKS’ THEOREM. Suppose that k ≧ 3 and that G is a k-regular graph which does not induce a (k + 1)-clique. Then G is k-colorable.Brooks proved his theorem in [1]; several more recent proofs have appeared in [3], [4] and [5].
James H. Schmerl
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A Fractional Analogue of Brooks' Theorem [PDF]
SIAM Journal on Discrete Mathematics, 2011 Third version, add Andrew King as an ...
Andrew D. King, Linyuan Lü, Xing Peng
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A Reconfigurations Analogue of Brooks' Theorem and Its Consequences [PDF]
Journal of Graph Theory, 2015 AbstractLet G be a simple undirected connected graph on n vertices with maximum degree Δ. Brooks' Theorem states that G has a proper Δ‐coloring unless G is a complete graph, or a cycle with an odd number of vertices. To recolor G is to obtain a new proper coloring by changing the color of one vertex.
Carl Feghali +2 more
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Brooks-type theorems for relaxations of square colorings [PDF]
Discrete Mathematics, 2023 The following relaxation of proper coloring the square of a graph was recently introduced: for a positive integer $h$, the proper $h$-conflict-free chromatic number of a graph $G$, denoted $χ_{pcf}^h(G)$, is the minimum $k$ such that $G$ has a proper $k$-coloring where every vertex $v$ has $\min\{deg_G(v),h\}$ colors appearing exactly once on its ...
Eun‐Kyung Cho +3 more
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Nondegenerate colourings in the Brooks theorem [PDF]
Discrete Mathematics and Applications, 2009 Let $c\geq 2$ and $p\geq c$ be two integers. We will call a proper coloring of the graph $G$ a \textit{$(c,p)$-nondegenerate}, if for any vertex of $G$ with degree at least $p$ there are at least $c$ vertices of different colors adjacent to it. In our work we prove the following result, which generalizes Brook's Theorem.
Nick Gravin
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A short proof of Brooks' theorem [PDF]
, 2018 We give a simple short proof of Brooks' theorem using only induction and greedy coloring, while avoiding issues of graph connectivity. The argument generalizes easily to some extensions of Brooks' theorem, including its variants for list coloring, signed graphs coloring and correspondence coloring.
Mariusz Zając
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Brooks' theorem for 2-fold coloring
Contributions to Discrete Mathematics, 2022 The two-fold chromatic number of a graph is the minimum number of colors needed to ensure that there is a way to color the graph so that each vertex gets two distinct colors, and adjacent vertices have no colors in common. The Ore degree is the maximum sum of degrees of an edge in a graph.
Jacob A. White
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A Reconfigurations Analogue of Brooks’ Theorem [PDF]
, 2014 Let G be a simple undirected graph on n vertices with maximum degree Δ. Brooks’ Theorem states that G has a Δ-colouring unless G is a complete graph, or a cycle with an odd number of vertices. To recolour G is to obtain a new proper colouring by changing the colour of one vertex.
Carl Feghali +2 more
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A Strengthening of Brooks' Theorem for Line Graphs [PDF]
The Electronic Journal of Combinatorics, 2011 We prove that if $G$ is the line graph of a multigraph, then the chromatic number $\chi(G)$ of $G$ is at most $\max\left\{\omega(G), \frac{7\Delta(G) + 10}{8}\right\}$ where $\omega(G)$ and $\Delta(G)$ are the clique number and the maximum degree of $G$, respectively.
Landon Rabern
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Strengthened Brooks Theorem for digraphs of girth three [PDF]
, 2011 12 ...
Ararat Harutyunyan, Bojan Mohar
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