Abstract
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.
Highlights
We prove that the gonality of a finite connected graph is lower bounded by its treewidth
In [2], Baker and Norine developed a theory of divisors on finite graphs analogous to divisor theory for Riemann surfaces
The gonality can be defined as the minimum degree of a positive rank divisor
Summary
The graphs in this paper will be finite and undirected (unless stated otherwise). We allow our graphs to have multiple (parallel) edges, but no loops. Let D0 and D be equivalent effective divisors satisfying D = D0. The set S is finite since the number of effective divisors equivalent to D is finite. Let D and D be two different, but equivalent effective divisors. The following lemma is similar to a theorem of Luo [16] on rank determining sets in the context of metric graphs. Let G = (V, E) be a graph (we allow multiple edges and loops). In order to use treewidth as a lower bound, we will use a characterisation of treewidth by Seymour and Thomas [19] in terms of “brambles”.(3) Let G = (V, E) be a graph, and let 2V denote the power set of V.
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