Related papers: The reverse mathematics of Brooks' theorem

We prove that for $k\geq 3$, the bound given by Brooks' theorem on the chromatic number of $k$-th powers of graphs of maximum degree $\Delta \geq 3$ can be lowered by 1, even in the case of online list coloring.

We consider extensions of Brooks' classic theorem on vertex coloring where some colors cannot be used on certain vertices. In particular we prove that if $G$ is a connected graph with maximum degree $\Delta(G) \geq 4$ that is not a complete…

Let $G$ be a simple undirected graph on $n$ vertices with maximum degree~$\Delta$. Brooks' Theorem states that $G$ has a $\Delta$-colouring unless~$G$ is a complete graph, or a cycle with an odd number of vertices. To recolour $G$ is to…

Coloring a graph $G$ consists in finding an assignment of colors $c: V(G)\to\{1,\ldots,p\}$ such that any pair of adjacent vertices receives different colors. The minimum integer $p$ such that a coloring exists is called the chromatic…

In the past, analogues to Brooks' theorem have been found for various parameters of graph coloring for infinite locally finite connected graphs in ZFC. We prove these theorems are not provable in ZF (i.e. the Zermelo-Fraenkel set theory…

We prove a descriptive version of Brooks's theorem for directed graphs. In particular, we show that, if $D$ is a Borel directed graph on a standard Borel space $X$ such that the maximum degree of each vertex is at most $d \geq 3$, then…

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…

K\"onig's edge coloring theorem says that a bipartite graph with maximal degree $n$ has an edge coloring with no more than $n$ colors. We explore the computability theory and Reverse Mathematics aspects of this theorem. Computable bipartite…

Let $\Delta(G)$ be the maximum degree of a graph $G$. Brooks' theorem states that the only connected graphs with chromatic number $\chi(G)=\Delta(G)+1$ are complete graphs and odd cycles. We prove a fractional analogue of Brooks' theorem in…

Working in subsystems of second order arithmetic, we formulate several representations for hypergraphs. We then prove the equivalence of various vertex coloring theorems to ${\sf WKL}_0$, ${\sf ACA}_0$ and $\Pi ^1_ 1$-${\sf CA}_0$.

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…

We generalize Brooks's theorem to show that if $G$ is a Borel graph on a standard Borel space $X$ of degree bounded by $d \geq 3$ which contains no $(d+1)$-cliques, then $G$ admits a $\mu$-measurable $d$-coloring with respect to any Borel…

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…

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 $\Delta$, has a vertex-colouring with $\Delta$ colours, unless $G$ is a…

We investigate the classical and distributed complexity of \emph{$k$-partial $c$-coloring} where $c=k$, a natural generalization of Brooks' theorem where each vertex should be colored from the palette $\{1,\ldots,c\} = \{1,\ldots,k\}$ such…

A vertex-coloring of a hypergraph is conflict-free, if each edge contains a vertex whose color is not repeated on any other vertex of that edge. Let $f(r, \Delta)$ be the smallest integer $k$ such that each $r$-uniform hypergraph of maximum…

Brooks' Theorem states that a connected graph $G$ of maximum degree $\Delta$ has chromatic number at most $\Delta$, unless $G$ is an odd cycle or a complete graph. A result of Johansson (1996) shows that if $G$ is triangle-free, then the…

We prove analogs of Brooks' Theorem for the list-distinguishing chromatic number of different classes of simple finite connected graphs. Moreover, we determine two upper bounds for the list-distinguishing chromatic number of a graph G in…

Circular $r$-coloring of a signed graph $(G,\sigma)$ is a mapping of its vertices to a circle of circumference $r$ such that: I. each pair of vertices with a negative connection is at distance at least $1$, and II. for each pair with a…

A graph is called to be uniquely list colorable, if it admits a list assignment which induces a unique list coloring. We study uniquely list colorable graphs with a restriction on the number of colors used. In this way we generalize a…