On metric graphs with prescribed gonality

Cools, Filip; Draisma, Jan

doi:10.1016/j.jcta.2017.11.017

Cited by 13 publications

(20 citation statements)

References 21 publications

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“…The gonality of a graph G is at most g(G)+3 2 . This conjecture has been confirmed for graphs with g(G) ≤ 5 in [AR18], with strong additional evidence coming from [CD18].…”

Section: Introductionmentioning

confidence: 76%

On the gonality of Cartesian products of graphs

Aidun¹,

Morrison²

2019

Preprint

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In this paper we study Cartesian products of graphs and their divisorial gonality, which is a tropical version of the gonality of an algebraic curve. We present an upper bound on the gonality of the Cartesian product of any two graphs, and provide instances where this bound holds with equality, including for the m × n rook's graph with min{m, n} ≤ 5. We use our upper bound to prove that Baker's gonality conjecture holds for the Cartesian product of any two graphs with two or more vertices each, and we determine precisely which nontrivial product graphs have gonality equal to Baker's conjectural upper bound.

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“…The gonality of a graph G is at most g(G)+3 2 . This conjecture has been confirmed for graphs with g(G) ≤ 5 in [AR18], with strong additional evidence coming from [CD18].…”

Section: Introductionmentioning

confidence: 76%

On the gonality of Cartesian products of graphs

Aidun¹,

Morrison²

2019

Preprint

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“…Part 2 of this conjecture has been proven by Cools, Draisma, Payne and Robeva [9]. Cools and Draisma [8] have also made significant progress on Part 1 of this conjecture. In answering Question 3 in Section 4 of this paper, we provide an alternative proof of part 2 of this conjecture.…”

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confidence: 82%

Sparse Graphs of High Gonality

Hendrey¹

2018

SIAM J. Discrete Math.

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By considering graphs as discrete analogues of Riemann surfaces, Baker and Norine (Adv. Math. 2007) developed a concept of linear systems of divisors for graphs. Building on this idea, a concept of gonality for graphs has been defined and has generated much recent interest. We show that there are connected graphs of treewidth 2 of arbitrarily high gonality. We also show that there exist pairs of connected graphs {G, H} such that H ⊆ G and H has strictly lower gonality than G. These results resolve three open problems posed in a recent survey by Norine (Surveys in Combinatorics 2015).

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“…The Duality Theorem still holds for tropical polynomials in three variables and the surfaces they define 15 . This time, instead of a Newton polygon we consider a New-ton polytope, the convex hull of all exponent vectors appearing in the polynomial.…”

Section: Tropical Surfaces and The Duality Theoremmentioning

confidence: 99%

Tropical Geometry

Morrison

2020

Foundations for Undergraduate Research in Mathematics

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Tropical mathematics redefines the rules of arithmetic by replacing addition with taking a maximum, and by replacing multiplication with addition. After briefly discussing a tropical version of linear algebra, we study polynomials build with these new operations. These equations define piecewise-linear geometric objects called tropical varieties. We explore these tropical varieties in two and three dimensions, building up discrete tools for studying them and determining their geometric properties. We then discuss the relationship between tropical geometry and algebraic geometry, which considers shapes defined by usual polynomial equations.Suggested prerequisites. We use standard set theory notation (unions, functions, etc.) throughout this chapter. Section 1 draws on terminology and motivation from abstract algebra and linear algebra, but can be understood without them. Section 2 draws on topics from discrete geometry, although it is mostly self-contained. Section 3 includes geometry in three dimensions, which uses some notation from a standard course in multivariable calculus. Section 4 uses ring theory terminology from an abstract algebra course.

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On metric graphs with prescribed gonality

Cited by 13 publications

References 21 publications

On the gonality of Cartesian products of graphs

On the gonality of Cartesian products of graphs

Sparse Graphs of High Gonality

Tropical Geometry

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