The 2-Factor Polynomial Detects Even Perfect Matchings

Baldridge, Scott; Lowrance, Adam M.; McCarty, Ben

doi:10.48550/arxiv.1812.10346

Cited by 2 publications

(9 citation statements)

References 15 publications

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“…Thus, if the conjecture is true, then the 2-factor polynomial must be zero for such a pair (G, M ) when evaluated at 1. This fact is indeed true, and it represents a significant step in proving Conjecture 1.2: Theorem 1.5 (Baldridge-Lowrance-McCarty [3]). Let G be a planar trivalent graph with perfect matching M .…”

Section: Introductionmentioning

confidence: 73%

“…Given an n-regular graph G and an ℓ-factor M of G, then a k-factor K factors through M if M is a subgraph of K. Denote the set of all k-factors of G that factor through M by 1 To the best of my knowledge, this idea is new in graph theory. This conjecture was proven in [3] after this paper was completed but before it was published.…”

Section: Introductionmentioning

confidence: 85%

“…A bridge (or isthmus or cut-edge) is an edge of a graph whose deletion increases the connected components of the resulting graph. 3 An edge is a bridge if and only if it is not contained in any cycle. The non-loop edges in the graph above are all bridges.…”

Section: Perfect Matchingsmentioning

confidence: 99%

“…However, invariants in graph theory tend to focus on counts of features of the graph and not necessarily the topology. We have already seen in Conjecture 1.2 (now a theorem, see [3]) that the 2-factor polynomial counts 2-factors. This implies, via Theorem 1.5 and Conjecture 1.6, that the 2-factor polynomial can detect whether a graph is 4-colorable.…”

Section: Introductionmentioning

confidence: 95%

“…Again, a surprise: the polynomial detected when a perfect matching M of G was even (cf. [3]). It seemed to do more, counting the number of 2-factors that factor through the 1-factor M , i.e., |G:M | 2 (see Conjecture 1.2).…”

Section: Introductionmentioning

confidence: 99%

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A Cohomology Theory for Planar Trivalent Graphs with Perfect Matchings

Baldridge¹

2018

Preprint

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In this paper, we prove a new cohomology theory that is an invariant of a planar trivalent graph with a given perfect matching. This bigraded cohomology theory appears to be very powerful: the graded Euler characteristic of the cohomology is a one variable polynomial (called the 2-factor polynomial) that, if nonzero when evaluated at one, implies that the perfect matching is even. This polynomial can be used to construct a polynomial invariant of the graph called the Tait polynomial. We conjecture that the Tait polynomial is positive when evaluated at one for all bridgeless planar trivalent graphs. This conjecture, if true, implies the existence of an even perfect matching for the graph, and thus the trivalent planar graph is 3-edge-colorable. This is equivalent to the four color theorem-a famous conjecture in mathematics that was proven with the aid of a computer in the 1970s. While these polynomial invariants may not have enough strength as invariants to prove such a conjecture directly, it is hoped that the strictly stronger cohomology theory developed in this paper will shed light on these types of problems.

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Section: Introductionmentioning

confidence: 73%

Section: Introductionmentioning

confidence: 85%

Section: Perfect Matchingsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

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A Cohomology Theory for Planar Trivalent Graphs with Perfect Matchings

Baldridge¹

2018

Preprint

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Unoriented Virtual Khovanov Homology

Baldridge,

Kauffman,

McCarty

2020

Preprint

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The Jones polynomial and Khovanov homology of a classical link are invariants that depend upon an initial choice of orientation for the link. In this paper, we define and verify a Khovanov homology theory for unoriented virtual links. (Virtual links include all classical links.) The graded Euler characteristic of this homology is proportional to a new unoriented Jones polynomial for virtual links, which is an invariant in the category of virtual links. The unoriented Jones polynomial continues to satisfy an important property of the usual (oriented) Jones polynomial: for classical or even links, the unoriented Jones polynomial evaluated at one is two to the power of the number of components of the link. As part of extending the main results of this paper to non-classical virtual links, a new, simpler definition for integral Khovanov homology is described. Finally, we define unoriented Lee homology theory for virtual links based upon the unoriented version of Khovanov homology. The Khovanov homology theory for virtual links defined here is a considerable simplification of the previous definitions due to Manturov and to Dye, Kaestner and Kauffman.

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The 2-Factor Polynomial Detects Even Perfect Matchings

Abstract: In this paper, we prove that the 2-factor polynomial, an invariant of a planar trivalent graph with a perfect matching, counts the number of 2factors that contain the the perfect matching as a subgraph. Consequently, we show that the polynomial detects even perfect matchings.

Cited by 2 publications

References 15 publications

A Cohomology Theory for Planar Trivalent Graphs with Perfect Matchings

A Cohomology Theory for Planar Trivalent Graphs with Perfect Matchings

Unoriented Virtual Khovanov Homology

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