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Landon Rabern

Publications and source records attributed to Landon Rabern.

At least 37 records · Page 2Linked to original sources
arXiv2015

The Fractional Chromatic Number of the Plane

The chromatic number of the plane is the chromatic number of the uncountably infinite graph that has as its vertices the points of the plane and has an edge between two points if their distance is 1. This chromatic number is denoted $χ(\mathcal{R}^2)$. The problem was introduced in 1950, and shortly thereafter it was proved that $4\le χ(\mathcal{R}^2)\le 7$. These bounds are both easy to prove, but after more than 60 years they are still the best known. In this paper, we investigate $χ_f(\mathcal{R}^2)$, the fractional chromatic number of the plane. The previous best bounds (rounded to five decimal places) were $3.5556 \le χ_f(\mathcal{R}^2)\le 4.3599$. Here we improve the lower bound to $76/21\approx3.6190$.

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arXiv2014

A note on coloring vertex-transitive graphs

We prove bounds on the chromatic number $χ$ of a vertex-transitive graph in terms of its clique number $ω$ and maximum degree $Δ$. We conjecture that every vertex-transitive graph satisfies $χ\le \max \left\{ω, \left\lceil\frac{5Δ+ 3}{6}\right\rceil\right\}$ and we prove results supporting this conjecture. Finally, for vertex-transitive graphs with $Δ\ge 13$ we prove the Borodin-Kostochka conjecture, i.e., $χ\le\max\{ω,Δ-1\}$.

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arXiv2014

Brooks' 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 analogues of Brooks' Theorem hold in each context. We conclude with two conjectures along the lines of Brooks' Theorem that are much stronger, the Borodin--Kostochka Conjecture and Reed's Conjecture.

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arXiv2014

Painting Squares in $Δ^2-1$ Shades

Cranston and Kim conjecture that if $G$ is a connected graph with maximum degree $Δ$ and $G$ is not a Moore Graph, then $χ_l(G^2) \le Δ^2-1$; here $χ_l$ is the list chromatic number. We prove their conjecture; in fact, this upper bound holds even for online list chromatic number.

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arXiv2013

A different short proof of Brooks' theorem

Lovász gave a short proof of Brooks' theorem by coloring greedily in a good order. We give a different short proof by reducing to the cubic case. Then we show how to extend the result to (online) list coloring via the Kernel Lemma.

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arXiv2012

A game generalizing Hall's theorem

We characterize the initial positions from which the first player has a winning strategy in a certain two-player game. This provides a generalization of Hall's theorem. Vizing's edge coloring theorem follows from a special case.

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arXiv2012

Coloring graphs with dense neighborhoods

It is shown that any graph with maximum degree $Δ$ in which the average degree of the induced subgraph on the set of all neighbors of any vertex exceeds $\frac{6k^2}{6k^2 + 1}Δ+ k + 6$ is either $(Δ- k)$-colorable or contains a clique on more than $Δ- 2k$ vertices. In the $k=1$ case we improve the bound on the average degree to $\frac23Δ+ 4$ and the bound on the clique number to $Δ-1$. As corollaries, we show that every graph satisfies $χ\leq \max\set{ω, Δ- 1, 4α}$ and every graph satisfies $χ\leq \max\set{ω, Δ- 1, \ceil{\frac{15 + \sqrt{48n + 73}}{4}}}$.

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arXiv2012

Partitioning and coloring with degree constraints

We prove that if $G$ is a vertex critical graph with $χ(G) \geq Δ(G) + 1 - p \geq 4$ for some $p \in \mathbb{N}$ and $ω(\fancy{H}(G)) \leq \frac{χ(G) + 1}{p + 1} - 2$, then $G = K_{χ(G)}$ or $G = O_5$. Here $\fancy{H}(G)$ is the subgraph of $G$ induced on the vertices of degree at least $χ(G)$. This simplifies and improves the results in the paper of Kostochka, Rabern and Stiebitz \cite{krs_one}.

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arXiv2011

An improvement on Brooks' Theorem

We prove that $χ(G) \leq \max {ω(G), Δ_2(G), (5/6)(Δ(G) + 1)}$ for every graph $G$ with $Δ(G) \geq 3$. Here $Δ_2$ 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.

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arXiv2011

A note on vertex partitions

We prove a general lemma about partitioning the vertex set of a graph into subgraphs of bounded degree. This lemma extends a sequence of results of Lovász, Catlin, Kostochka and Rabern.

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arXiv2011

Destroying Non-Complete Regular Components in Graph Partitions

We prove that if $G$ is a graph and $r_1, ..., r_k \in \mathbb{Z}_{\geq 0}$ such that $\sum_{i=1}^k r_i \geq Δ(G) + 2 - k$ then $V(G)$ can be partitioned into sets $V_1, ..., V_k$ such that $Δ(G[V_i]) \leq r_i$ and $G[V_i]$ contains no non-complete $r_i$-regular components for each $1 \leq i \leq k$. In particular, the vertex set of any graph $G$ can be partitioned into $\left \lceil \frac{Δ(G) + 2}{3} \right \rceil$ sets, each of which induces a disjoint union of triangles and paths.

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arXiv2011

Coloring $Δ$-Critical Graphs With Small High Vertex Cliques

We prove that $K_{χ(G)}$ is the only critical graph $G$ with $χ(G) \geq Δ(G) \geq 6$ and $ω(\mathcal{H}(G)) \leq \left \lfloor \frac{Δ(G)}{2} \right \rfloor - 2$. Here $\mathcal{H}(G)$ is the subgraph of $G$ induced on the vertices of degree at least $χ(G)$. Setting $ω(\mathcal{H}(G)) = 1$ proves a conjecture of Kierstead and Kostochka.

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arXiv2010

On hitting all maximum cliques with an independent set

We prove that every graph $G$ for which $ω(G) \geq 3/4(Δ(G) + 1)$, has an independent set $I$ such that $ω(G - I) < ω(G)$. It follows that a minimum counterexample $G$ to Reed's conjecture satisfies $ω(G) < 3/4(Δ(G) + 1)$ and hence also $χ(G) > \lceil 7/6ω(G) \rceil$. We also prove that if for every induced subgraph $H$ of $G$ we have $χ(H) \leq \max{\lceil 7/6ω(H) \rceil, \lceil \frac{ω(H) + Δ(H) + 1}{2}\rceil}$, then we also have $χ(G) \leq \lceil \frac{ω(G) + Δ(G) + 1}{2}\rceil$. This gives a generic proof of the upper bound for line graphs of multigraphs proved by King et al.

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arXiv2008

A Novel Proof of the Heine-Borel Theorem

Every beginning real analysis student learns the classic Heine-Borel theorem, that the interval [0,1] is compact. In this article, we present a proof of this result that doesn't involve the standard techniques such as constructing a sequence and appealing to the completeness of the reals. We put a metric on the space of infinite binary sequences and prove that compactness of this space follows from a simple combinatorial lemma. The Heine-Borel theorem is an immediate corollary.

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arXiv2007

Coloring and The Lonely Graph

We improve upper bounds on the chromatic number proven independently in \cite{reedNote} and \cite{ingo}. Our main lemma gives a sufficient condition for two paths in graph to be completely joined. Using this, we prove that if a graph has an optimal coloring with more than $\fracω{2}$ singleton color classes, then it satisfies $χ\leq \frac{ω+ Δ+ 1}{2}$. It follows that a graph satisfying $n - Δ< α+ \frac{ω- 1}{2}$ must also satisfy $χ\leq \frac{ω+ Δ+ 1}{2}$, improving the bounds in \cite{reedNote} and \cite{ingo}. We then give a simple argument showing that if a graph satisfies $χ> \frac{n + 3 - α}{2}$, then it also satisfies $χ(G) \leq \left\lceil\frac{ω(G) + Δ(G) + 1}{2}\right\rceil$. From this it follows that a graph satisfying $n - Δ< α+ ω$ also satisfies $χ(G) \leq \left\lceil\frac{ω(G) + Δ(G) + 1}{2}\right\rceil$ improving the bounds in \cite{reedNote} and \cite{ingo} even further at the cost of a ceiling. In the next sections, we generalize our main lemma to constrained colorings (e.g. r-bounded colorings). We present a generalization of Reed's conjecture to r-bounded colorings and prove the conjecture for graphs with maximal degree close to their order. Finally, we outline some applications (in \cite{BorodinKostochka} and \cite{ColoringWithDoublyCriticalEdge}) of the theory presented here to the Borodin-Kostochka conjecture and coloring graphs containing a doubly critical edge.

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