Catalan-many tropical morphisms to trees; Part I: Constructions

Discrete and metric divisorial gonality can be different

“…3.12] using algebraic geometry. A purely combinatorial proof of this result was recently found by Draisma and Vargas [14], with many promising avenues still to be explored [15]. However, for discrete graphs, Conjecture 1.1 is still wide open.…”

“…In [3], Baker posed a number of open problems in the theory of divisors on graphs. All but two of these have since been solved; see [19,20,10,14]. The first and most important remaining open problem is the Brill-Noether conjecture for finite graphs [3,Conj.…”

On the Gonality of Metric Graphs

“…The upper bound on the gonality of metric graphs is a purely combinatorial statement whose proof, up to now, depended on algebraic geometry. We have recently completed a purely combinatorial construction of maps witnessing the gonality bound [DV21,Var20]. The technical details of our construction are somewhat daunting-and indeed, we hope that this exposition might incite interest that leads to a simplification!-but the key ideas are very simple, and we discuss these in Section 4.…”

Complexity of Graph Problems: Gonality, Colouring and Scheduling

“…Furthermore, through methods of algebraic geometry the upper bound 1 +3 2 has been established for stable gonality, where 1 ( ) is the first Betti number, also known as cyclomatic number or circuit rank, and equals | ( )| − | ( )| + 1 [35]. A combinatorial proof for the metric variant is given in [44], and by considering a graph as a metric graph with edge lengths 1, this is a combinatorial proof for (non-metric) graphs as well (a combinatorial proof for (metric) graphs with maximum degree 3 was already given in [32]). It has been conjectured that 1 +3…”