A Note on Brill–Noether Existence for Graphs of Low Genus

Atanasov, Stanislav; Ranganathan, Dhruv

doi:10.1307/mmj/1519095622

Cited by 7 publications

(5 citation statements)

References 20 publications

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“…However, for discrete graphs, Conjecture 1.1 is still wide open. 1 Partial results were obtained by Atanasov and Ranganathan [2], who proved Conjecture 1.1 for all graphs of genus at most 5, and by Aidun and Morrison [1], who proved the conjecture for Cartesian product graphs.…”

Section: Introductionmentioning

confidence: 99%

Discrete and metric divisorial gonality can be different

Bruyn

Smit

Wegen

2022

Journal of Combinatorial Theory, Series A

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Section: Introductionmentioning

confidence: 99%

Discrete and metric divisorial gonality can be different

Bruyn

Smit

Wegen

2022

Journal of Combinatorial Theory, Series A

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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: 75%

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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“…Recall Conjecture 1.1, which says that if G is a graph of genus g and ρ(g, r, d) ≥ 0, then gon r (G) ≤ d. (Here ρ(g, r, d) := g − (r + 1)(g − d + r).) It is proven in [2] that this conjecture holds for all graphs of genus g ≤ 5. Hence, we have the following result, which will be useful in Section 4.…”

Section: Introductionmentioning

confidence: 95%

“…In the language of higher gonalities, this conjecture can be rephrased as saying that if G is a graph of genus g and we have ρ(g, r, d) ≥ 0, then gon r (G) ≤ d. This conjecture has been verified for graphs of genus at most 5 [2]. In the special case of r = 1, the conjecture is an upper bound on the first gonality of a graph.…”

Section: Introductionmentioning

confidence: 98%

Gonality sequences of graphs

Aidun¹,

Dean²,

Morrison³

et al. 2020

Preprint

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To any graph we associate a sequence of integers called the gonality sequence of the graph, consisting of the minimum degrees of divisors of increasing rank on the graph. This is a tropical analogue of the gonality sequence of an algebraic curve. We study gonality sequences for graphs of low genus, proving that for genus up to 5, the gonality sequence is determined by the genus and the first gonality. We then prove that any reasonable pair of first two gonalities is achieved by some graph. We also develop a modified version of Dhar's burning algorithm more suited for studying higher gonalities.

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A Note on Brill–Noether Existence for Graphs of Low Genus

Cited by 7 publications

References 20 publications

Discrete and metric divisorial gonality can be different

Discrete and metric divisorial gonality can be different

On the gonality of Cartesian products of graphs

Gonality sequences of graphs

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