Abstract: It is shown that any graph with maximum degree $\ Delta $ in which the average degree o... more Abstract: It is shown that any graph with maximum degree $\ Delta $ in which the average degree of the induced subgraph on the set of all neighbors of any vertex exceeds $\ frac {6k^ 2}{6k^ 2+ 1}\ Delta+ k+ 6$ is either $(\ Delta-k) $-colorable or contains a clique on more than $\ Delta-2k $ vertices. In the $ k= 1$ case we improve the bound on the average degree to $\ frac23\ Delta+ 4$ and the bound on the clique number to $\ Delta-1$.
Abstract: Every beginning real analysis student learns the classic Heine-Borel theorem, that the ... more Abstract: 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.
We prove that every claw-free graph G that doesn't contain a clique on ∆(G) ≥ 9 vertices can be ∆... more We prove that every claw-free graph G that doesn't contain a clique on ∆(G) ≥ 9 vertices can be ∆(G) − 1 colored.
We prove that every claw-free graph G that doesn't contain a clique on ∆(G) ≥ 9 vertices can be ∆... more We prove that every claw-free graph G that doesn't contain a clique on ∆(G) ≥ 9 vertices can be ∆(G) − 1 colored.
Lovász gave a short proof of Brooks' theorem by coloring greedily in a good order. We give a diff... more 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.
Borodin and Kostochka conjectured that every graph G with maximum degree ∆ ≥ 9 satisfies χ ≤ max ... more Borodin and Kostochka conjectured that every graph G with maximum degree ∆ ≥ 9 satisfies χ ≤ max {ω, ∆ − 1}. We carry out an in-depth study of minimum counterexamples to the Borodin-Kostochka conjecture. Our main tool is the classi-
We characterize the initial positions from which the first player has a winning strategy in a cer... more 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.
Let G be a color-critical graph with χ(G) ≥ Δ(G) = 2t + 1 ≥ 5 such that the subgraph of G induced... more Let G be a color-critical graph with χ(G) ≥ Δ(G) = 2t + 1 ≥ 5 such that the subgraph of G induced by the vertices of degree 2t + 1 has clique number at most t − 1. We prove that then either t ≥ 3 and G = K 2t+2 or t = 2 and G ∈ {K 6 , O 5 }, where O 5 is a special graph with χ(O 5 ) = 5 and |O 5 | = 9. This result for t ≥ 3 improves a case of a theorem by Rabern [9] and for t = 2 answers a question raised by Kierstead and Kostochka in [6].
The semantic paradoxes are often associated with self-reference or referential circularity. Yablo... more The semantic paradoxes are often associated with self-reference or referential circularity. Yablo [1993], however, has shown that there are infinitary versions of the paradoxes 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. In this essay, we lay the groundwork for a general investigation into the nature of reference structures that support the semantic paradoxes and the semantic hypodoxes. We develop a functionally complete infinitary propositional language endowed with a denotation assignment and extract the reference structural information in terms of graph-theoretic properties. We introduce the new concepts of dangerous and precarious reference graphs, which allows us to rigorously define the task: classify the dangerous and precarious directed graphs purely in terms of their graph-theoretic properties. Ungroundedness will be shown to fully characterize the precarious reference graphs and fully characterize the dangerous finite graphs. We prove that an undirected graph has a dangerous orientation if and only if it contains a cycle, providing some support for the traditional idea that cyclic structure is required for paradoxicality. This leaves the task of classifying danger for infinite acyclic reference graphs. We provide some compactness results, which give further necessary conditions on danger in infinite graphs, which in conjunction with a notion of self-containment allows us to prove that dangerous acyclic graphs must have infinitely many vertices with infinite out-degree. But a full characterization of danger remains an open question. In the appendices we relate our results to the results given in Cook [2004] and Yablo [2006] with respect to more restricted sentences systems, which we call F-systems. 1 An early version of Yablo's ω-paradox can be found in Yablo [1985], p. 340.
We prove a general lemma about partitioning the vertex set of a graph into subgraphs of bounded d... more 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 Lovasz, Catlin, Kostochka and Rabern.
Abstract: It is shown that any graph with maximum degree $\ Delta $ in which the average degree o... more Abstract: It is shown that any graph with maximum degree $\ Delta $ in which the average degree of the induced subgraph on the set of all neighbors of any vertex exceeds $\ frac {6k^ 2}{6k^ 2+ 1}\ Delta+ k+ 6$ is either $(\ Delta-k) $-colorable or contains a clique on more than $\ Delta-2k $ vertices. In the $ k= 1$ case we improve the bound on the average degree to $\ frac23\ Delta+ 4$ and the bound on the clique number to $\ Delta-1$.
Abstract: Every beginning real analysis student learns the classic Heine-Borel theorem, that the ... more Abstract: 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.
We prove that every claw-free graph G that doesn't contain a clique on ∆(G) ≥ 9 vertices can be ∆... more We prove that every claw-free graph G that doesn't contain a clique on ∆(G) ≥ 9 vertices can be ∆(G) − 1 colored.
We prove that every claw-free graph G that doesn't contain a clique on ∆(G) ≥ 9 vertices can be ∆... more We prove that every claw-free graph G that doesn't contain a clique on ∆(G) ≥ 9 vertices can be ∆(G) − 1 colored.
Lovász gave a short proof of Brooks' theorem by coloring greedily in a good order. We give a diff... more 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.
Borodin and Kostochka conjectured that every graph G with maximum degree ∆ ≥ 9 satisfies χ ≤ max ... more Borodin and Kostochka conjectured that every graph G with maximum degree ∆ ≥ 9 satisfies χ ≤ max {ω, ∆ − 1}. We carry out an in-depth study of minimum counterexamples to the Borodin-Kostochka conjecture. Our main tool is the classi-
We characterize the initial positions from which the first player has a winning strategy in a cer... more 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.
Let G be a color-critical graph with χ(G) ≥ Δ(G) = 2t + 1 ≥ 5 such that the subgraph of G induced... more Let G be a color-critical graph with χ(G) ≥ Δ(G) = 2t + 1 ≥ 5 such that the subgraph of G induced by the vertices of degree 2t + 1 has clique number at most t − 1. We prove that then either t ≥ 3 and G = K 2t+2 or t = 2 and G ∈ {K 6 , O 5 }, where O 5 is a special graph with χ(O 5 ) = 5 and |O 5 | = 9. This result for t ≥ 3 improves a case of a theorem by Rabern [9] and for t = 2 answers a question raised by Kierstead and Kostochka in [6].
The semantic paradoxes are often associated with self-reference or referential circularity. Yablo... more The semantic paradoxes are often associated with self-reference or referential circularity. Yablo [1993], however, has shown that there are infinitary versions of the paradoxes 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. In this essay, we lay the groundwork for a general investigation into the nature of reference structures that support the semantic paradoxes and the semantic hypodoxes. We develop a functionally complete infinitary propositional language endowed with a denotation assignment and extract the reference structural information in terms of graph-theoretic properties. We introduce the new concepts of dangerous and precarious reference graphs, which allows us to rigorously define the task: classify the dangerous and precarious directed graphs purely in terms of their graph-theoretic properties. Ungroundedness will be shown to fully characterize the precarious reference graphs and fully characterize the dangerous finite graphs. We prove that an undirected graph has a dangerous orientation if and only if it contains a cycle, providing some support for the traditional idea that cyclic structure is required for paradoxicality. This leaves the task of classifying danger for infinite acyclic reference graphs. We provide some compactness results, which give further necessary conditions on danger in infinite graphs, which in conjunction with a notion of self-containment allows us to prove that dangerous acyclic graphs must have infinitely many vertices with infinite out-degree. But a full characterization of danger remains an open question. In the appendices we relate our results to the results given in Cook [2004] and Yablo [2006] with respect to more restricted sentences systems, which we call F-systems. 1 An early version of Yablo's ω-paradox can be found in Yablo [1985], p. 340.
We prove a general lemma about partitioning the vertex set of a graph into subgraphs of bounded d... more 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 Lovasz, Catlin, Kostochka and Rabern.
Brooks' Theorem from 1941 upper-bounds the chromatic number of a graph by the maximum of its cliq... more Brooks' Theorem from 1941 upper-bounds the chromatic number of a graph by the maximum of its clique number and max degree. In 2009, Kierstead and Kostochka proved a similar bound with the max degree replaced by half the Ore degree and conjectured the optimal bound of this form. We discuss the proof of this conjecture and some generalizations. It turns out that for each 0 < \epsilon \leq 1, there is an upper bound of the Brooks' form with max degree replaced by the maximum over all edges of a certain \epsilon-weighted average of the degrees of the ends. Additionally, we prove that if such a bound exists for \epsilon = 0, then P=NP.
Some of this is joint work with Kostochka and Stiebitz.
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Papers by Landon Rabern
Some of this is joint work with Kostochka and Stiebitz.