We prove that the (divisorial) gonality of a finite connected graph is lower bounded by its treewidth. Graphs for which equality holds include the grid graphs and the complete multipartite graphs. We prove that the treewidth lower bound also holds for metric graphs (tropical curves) by constructing for any positive rank divisor on a metric graph a positive rank divisor of the same degree on a subdivision of the underlying combinatorial graph. Finally, we show that the treewidth lower bound also holds for a related notion of gonality defined by Caporaso and for stable gonality as introduced by Cornelissen et al.

Revised: 2020-04-11

Accepted: 2020-04-12

Published online: 2020-08-20

Classification: 05C57, 05C83, 14T05, 14H51

Keywords: Gonality, treewidth, tropical curve, metric graph.

@article{ALCO_2020__3_4_941_0, author = {van Dobben de Bruyn, Josse and Gijswijt, Dion}, title = {Treewidth is a lower bound on graph~gonality}, journal = {Algebraic Combinatorics}, pages = {941--953}, publisher = {MathOA foundation}, volume = {3}, number = {4}, year = {2020}, doi = {10.5802/alco.124}, language = {en}, url = {alco.centre-mersenne.org/item/ALCO_2020__3_4_941_0/} }

van Dobben de Bruyn, Josse; Gijswijt, Dion. Treewidth is a lower bound on graph gonality. Algebraic Combinatorics, Volume 3 (2020) no. 4, pp. 941-953. doi : 10.5802/alco.124. https://alco.centre-mersenne.org/item/ALCO_2020__3_4_941_0/

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