Playing cards with Vizing's demon
We analyze a solitaire game in which a demon rearranges some cards after each move. The graph edge coloring theorems of Kőnig (1931) and Vizing (1964) follow from the winning strategies developed.
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arXiv subjects
Landon Rabern
Publications and source records attributed to Landon Rabern.
Structural fixed-point theorems
The semantic paradoxes are associated with self-reference or referential circularity. However, there are infinitary versions of the paradoxes, such as Yablo's paradox, that do not involve this form of circularity. It remains an open question what relations of reference between collections of sentences afford the structure necessary for paradoxicality -- these are the so-called "dangerous" directed graphs. Building on Rabern, et. al (2013) we reformulate this problem in terms of fixed points of certain functions, thereby boiling it down to get a purely mathematical problem.
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Coloring $(P_5, \text{gem})$-free graphs with $\Delta -1$ colors
The Borodin-Kostochka Conjecture states that for a graph $G$, if $\Delta(G) \geq 9$ and $\omega(G) \leq \Delta(G)-1$, then $\chi(G)\leq\Delta(G) -1$. We prove the Borodin-Kostochka Conjecture for $(P_5, \text{gem})$-free graphs, i.e., graphs with no induced $P_5$ and no induced $K_1\vee P_4$.
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Coloring a graph with $Δ-1$ colors: Conjectures equivalent to the Borodin-Kostochka conjecture that appear weaker
Borodin and Kostochka conjectured that every graph $G$ with maximum degree $Δ\ge 9$ satisfies $χ\le \max\{ω, Δ-1\}$. We carry out an in-depth study of minimum counterexamples to the Borodin-Kostochka conjecture. Our main tool is the identification of graph joins that are $f$-choosable, where $f(v) = d(v) - 1$ for each vertex $v$. Since such a join cannot be an induced subgraph of a vertex critical graph with $χ= Δ$, we have a wealth of structural information about minimum counterexamples to the Borodin-Kostochka conjecture. Our main result proves that certain conjectures that are prima facie weaker than the Borodin-Kostochka conjecture are in fact equivalent to it. One such equivalent conjecture is the following: Any graph with $χ\ge Δ= 9$ contains $K_3 * \bar{K_6}$ as a subgraph.
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The Hilton--Zhao Conjecture is True for Graphs with Maximum Degree 4
A simple graph $G$ is \emph{overfull} if $|E(G)|>Δ\lfloor|V(G)|/2\rfloor$. By the pigeonhole principle, every overfull graph $G$ has $χ'(G)>Δ$. The \emph{core} of a graph, denoted $G_Δ$, is the subgraph induced by its vertices of degree $Δ$. Vizing's Adjacency Lemma implies that if $χ'(G)>Δ$, then $G_Δ$ contains cycles. Hilton and Zhao conjectured that if $G_Δ$ has maximum degree 2 and $Δ\ge 4$, then $χ'(G)>Δ$ precisely when $G$ is overfull. We prove this conjecture for the case $Δ=4$.
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Subcubic edge chromatic critical graphs have many edges
We consider graphs $G$ with $Δ=3$ such that $χ'(G)=4$ and $χ'(G-e)=3$ for every edge $e$, so-called \emph{critical} graphs. Jakobsen noted that the Petersen graph with a vertex deleted, $P^*$, is such a graph and has average degree only $\frac83$. He showed that every critical graph has average degree at least $\frac83$, and asked if $P^*$ is the only graph where equality holds. A result of Cariolaro and Cariolaro shows that this is true. We strengthen this average degree bound further. Our main result is that if $G$ is a subcubic critical graph other than $P^*$, then $G$ has average degree at least $\frac{46}{17}\approx2.706$. This bound is best possible, as shown by the Hajos join of two copies of $P^*$.
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Short fans and the 5/6 bound for line graphs
In 2011, the second author conjectured that every line graph $G$ satisfies $χ(G)\le \max\{ω(G),\frac{5Δ(G)+8}{6}\}$. This conjecture is best possible, as shown by replacing each edge in a 5-cycle by $k$ parallel edges, and taking the line graph. In this paper we prove the conjecture. We also develop more general techniques and results that will likely be of independent interest, due to their use in attacking the Goldberg--Seymour conjecture.
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Planar graphs are 9/2-colorable
We show that every planar graph $G$ has a 2-fold 9-coloring. In particular, this implies that $G$ has fractional chromatic number at most $\frac92$. This is the first proof (independent of the 4 Color Theorem) that there exists a constant $k<5$ such that every planar $G$ has fractional chromatic number at most $k$.
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Planar Graphs have Independence Ratio at least 3/13
The 4 Color Theorem (4CT) implies that every $n$-vertex planar graph has an independent set of size at least $\frac{n}4$; this is best possible, as shown by the disjoint union of many copies of $K_4$. In 1968, Erdős asked whether this bound on independence number could be proved more easily than the full 4CT. In 1976 Albertson showed (independently of the 4CT) that every $n$-vertex planar graph has an independent set of size at least $\frac{2n}9$. Until now, this remained the best bound independent of the 4CT. Our main result improves this bound to $\frac{3n}{13}$.
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A better lower bound on average degree of k-list-critical graphs
We improve the best known bounds on average degree of $k$-list-critical graphs for $k \ge 6$. Specifically, for $k \ge 7$ we show that every non-complete $k$-list-critical graph has average degree at least $k-1 + \frac{(k-3)^2 (2 k-3)}{k^4-2 k^3-11 k^2+28 k-14}$ and every non-complete $6$-list-critical graph has average degree at least $5 + \frac{93}{766}$. The same bounds hold for online $k$-list-critical graphs.
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A better lower bound on average degree of 4-list-critical graphs
This short note proves that every incomplete $k$-list-critical graph has average degree at least $k-1 + \frac{k-3}{k^2-2k+2}$. This improves the best known bound for $k = 4,5,6$. The same bound holds for online $k$-list-critical graphs.
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Edge Lower Bounds for List Critical Graphs, via Discharging
A graph $G$ is $k$-critical if $G$ is not $(k-1)$-colorable, but every proper subgraph of $G$ is $(k-1)$-colorable. A graph $G$ is $k$-choosable if $G$ has an $L$-coloring from every list assignment $L$ with $|L(v)|=k$ for all $v$, and a graph $G$ is \emph{$k$-list-critical} if $G$ is not $(k-1)$-choosable, but every proper subgraph of $G$ is $(k-1)$-choosable. The problem of bounding (from below) the number of edges in a $k$-critical graph has been widely studied, starting with work of Gallai and culminating with the seminal results of Kostochka and Yancey, who essentially solved the problem. In this paper, we improve the best lower bound on the number of edges in a $k$-list-critical graph. Our proof uses the discharging method, which makes it simpler and more modular than previous work in this area.
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Improved lower bounds on the number of edges in list critical and online list critical graphs
We prove that every $k$-list-critical graph ($k \ge 7$) on $n \ge k+2$ vertices has at least $\frac12 \left(k-1 + \frac{k-3}{(k-c)(k-1) + k-3}\right)n$ edges where $c = (k-3)\left(\frac12 - \frac{1}{(k-1)(k-2)}\right)$. This improves the bound established by Kostochka and Stiebitz. The same bound holds for online $k$-list-critical graphs, improving the bound established by Riasat and Schauz. Both bounds follow from a more general result stating that either a graph has many edges or it has an Alon-Tarsi orientable induced subgraph satisfying a certain degree condition.
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Extracting list colorings from large independent sets
We take an application of the Kernel Lemma by Kostochka and Yancey to its logical conclusion. The consequence is a sort of magical way to draw conclusions about list coloring (and online list coloring) just from the existence of an independent set incident to many edges. We use this to prove an Ore-degree version of Brooks' Theorem for online list-coloring. The Ore-degree of an edge $xy$ in a graph $G$ is $θ(xy) = d_G(x) + d_G(y)$. The Ore-degree of $G$ is $θ(G) = \max_{xy\in E(G)}θ(xy)$. We show that every graph with $θ\ge18$ and $ω\le\fracθ{2}$ is online $\left\lfloor \fracθ{2}\right\rfloor $-choosable. In addition, we prove an upper bound for online list-coloring triangle-free graphs: $χ_{OL}\leΔ+1-\lfloor\frac{1}{4}\lg(Δ)\rfloor$. Finally, we characterize Gallai trees as the connected graphs $G$ with no independent set incident to at least $|G|$ edges.
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Beyond Degree Choosability
Let $G$ be a connected graph with maximum degree $Δ$. Brooks' theorem states that $G$ has a $Δ$-coloring unless $G$ is a complete graph or an odd cycle. A graph $G$ is \emph{degree-choosable} if $G$ can be properly colored from its lists whenever each vertex $v$ gets a list of $d(v)$ colors. In the context of list coloring, Brooks' theorem can be strengthened to the following. Every connected graph $G$ is degree-choosable unless each block of $G$ is a complete graph or an odd cycle; such a graph $G$ is a \emph{Gallai tree}. This degree-choosability result was further strengthened to Alon--Tarsi orientations; these are orientations of $G$ in which the number of spanning Eulerian subgraphs with an even number of edges differs from the number with an odd number of edges. A graph $G$ is \emph{degree-AT} if $G$ has an Alon--Tarsi orientation in which each vertex has indegree at least 1. Alon and Tarsi showed that if $G$ is degree-AT, then $G$ is also degree-choosable. Hladky, Kral, and Schauz showed that a connected graph is degree-AT if and only if it is not a Gallai tree. In this paper, we consider pairs $(G,x)$ where $G$ is a connected graph and $x$ is some specified vertex in $V(G)$. We characterize pairs such that $G$ has no Alon--Tarsi orientation in which each vertex has indegree at least 1 and $x$ has indegree at least 2. When $G$ is 2-connected, the characterization is simple to state.
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List-coloring claw-free graphs with $Δ$-1 colors
We prove that if $G$ is a quasi-line graph with $Δ(G)>ω(G)$ and $Δ(G)\ge 69$, then $χ_{OL}(G)\le Δ(G)-1$. Together with our previous work, this implies that if $G$ is a claw-free graph with $Δ(G)>ω(G)$ and $Δ(G)\ge 69$, then $χ_{\ell}(G)\le Δ(G)-1$.
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Graphs with $χ=Δ$ have big cliques
Brooks' Theorem states that if a graph has $Δ\ge 3$ and $ω\le Δ$, then $χ\le Δ$. Borodin and Kostochka conjectured that if $Δ\ge 9$ and $ω\le Δ-1$, then $χ\le Δ-1$. We show that if $Δ\ge 13$ and $ω\le Δ-4$, then $χ\le Δ-1$. For a graph $G$, let $\mathcal{H}$ denote the subgraph of $G$ induced by vertices of degree $Δ$. We also show that if $ω\le Δ-1$ and $ω(\mathcal{H})\le Δ-6$, then $χ\le Δ-1$.
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Edge-coloring via fixable subgraphs
Many graph coloring proofs proceed by showing that a minimal counterexample to the theorem being proved cannot contain certain configurations, and then showing that each graph under consideration contains at least one such configuration; these configurations are called \emph{reducible} for that theorem. (A \emph{configuration} is a subgraph $H$, along with specified degrees $d_G(v)$ in the original graph $G$ for each vertex of $H$.) We give a general framework for showing that configurations are reducible for edge-coloring. A particular form of reducibility, called \emph{fixability}, can be considered without reference to a containing graph. This has two key benefits: (i) we can now formulate necessary conditions for fixability, and (ii) the problem of fixability is easy for a computer to solve. The necessary condition of \emph{superabundance} is sufficient for multistars and we conjecture that it is sufficient for trees as well, which would generalize the powerful technique of Tashkinov trees. Via computer, we can generate thousands of reducible configurations, but we have short proofs for only a small fraction of these. The computer can write \LaTeX\ code for its proofs, but they are only marginally enlightening and can run thousands of pages long. We give examples of how to use some of these reducible configurations to prove conjectures on edge-coloring for small maximum degree. Our aims in writing this paper are (i) to provide a common context for a variety of reducible configurations for edge-coloring and (ii) to spur development of methods for humans to understand what the computer already knows.
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