Discrete and metric divisorial gonality can be different

This paper compares the divisorial gonality of a finite graph $G$ to the divisorial gonality of the associated metric graph $\Gamma(G,\mathbb{1})$ with unit lengths. We show that $\text{dgon}(\Gamma(G,\mathbb{1}))$ is equal to the minimal divisorial gonality of all regular subdivisions of $G$, and we provide a class of graphs for which this number is strictly smaller than the divisorial gonality of $G$. This settles a conjecture of M. Baker in the negative.


Introduction
Over the past 15 years, fruitful analogies between graphs and algebraic curves have been established, building on the seminal paper of Baker and Norine [5].In that paper, the authors proved a version of the Riemann-Roch theorem for divisors on a finite graph, and showed that their result is closely related to the combinatorial theory of chip-firing games (e.g.[7,6]).The paper of Baker and Norine, together with another paper by Baker [3], have led to a flurry of research into the interplay between graphs and curves, leading to new problems and results in combinatorics and to new combinatorial techniques in geometry (e.g. a combinatorial proof of the Brill-Noether theorem, [10]).
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. 3.9(1)], based on an analogous result for curves.We focus on the r = 1 case of this conjecture.Let dgon r (G) denote the smallest degree of a rank r divisor on G, and let dgon(G) := dgon 1 (G) denote the divisorial gonality of G (see §2 for definitions).Then the r = 1 case of the Brill-Noether conjecture can be stated as follows.The corresponding result for metric graphs was proven by Baker [3,Thm. 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. 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.
The most straightforward approach to Conjecture 1.1 would be to show that the divisorial gonality of a graph is equal to the divisorial gonality of the associated metric graph with unit lengths.This is the second remaining conjecture of Baker's paper [3,Conj. 3.14].Given a multigraph G and an integer k ≥ 1, let σ k (G) denote the multigraph obtained from G by subdividing every edge into k parts.The conjecture can then be stated as follows.
Conjecture 1.2 ([3, Conjecture 3.14]).Let G be a connected loopless multigraph, let Γ(G) be the corresponding metric graph with unit edge lengths, and let r ≥ 1. Then: A partial result in this direction was already implicit in the work of Gathmann and Kerber [17,Prop. 3.1] (see Theorem 3.3(a) below), but to our knowledge Theorem 1.3 is new.Moreover, we use a different proof technique, which can be used to give an upper bound on the number of subdivisions needed to get equality (see Remark 3.7).
The proof runs roughly as follows.It is already known that every rank r divisor on σ k (G) also defines a rank r divisor on Γ(G).For the converse, we show that every rank r divisor D on Γ(G) can be "rounded" to a nearby divisor D with rank(D ) ≥ r which is supported on the Q-points of Γ(G), and therefore on the points of some regular subdivision σ k (G).The details will be given in §3.
As pointed out by Baker in [3], a positive answer to Conjecture 1.2 would also yield a positive answer to Conjecture 1.1.However, it turns out that the subdivision conjecture fails, and we give a counterexample to Conjecture 1.2(a) in the case r = 1 and k = 2. Evidently this is also a counterexample to Conjecture 1.2(b).The second main result of this paper is the following.Theorem 1.4.For every integer k ≥ 1, there exists a connected loopless multigraph G k such that dgon(G k ) = 6k and dgon(Γ(G k )) = dgon(σ 2 (G k )) = 5k.Furthermore, G k can be chosen simple and bipartite.
The proof is constructive and consists of two parts.In §4, we construct a family of graphs with dgon(G) = 6 and dgon(Γ(G)) = dgon(σ 2 (G)) = 5.The graphs G k are then constructed in §5 by combining k of these graphs in a certain way.
Although the difference between dgon(G) and dgon(Γ(G)) can be large, as in Theorem 1.4, the ratio between them is at most 2, as we show in Proposition 5.3.Hence, for the gap to get arbitrarily large, it is necessary that dgon(Γ(G)) goes to infinity.
In §6, we list a few additional counterexamples (without proof), including a 3-regular graph.Although all counterexamples in this paper violate Conjecture 1.2, they nevertheless satisfy the Brill-Noether bound.We do not know whether any of these examples can be extended to disprove Conjecture 1.1.Additional open problems are discussed in §6 as well.

Preliminaries
Throughout this paper, a graph will be a finite, connected, loopless multigraph.In other words, parallel edges are allowed, but self-loops are not.

Divisors on graphs
A divisor on a graph G is an element of the free abelian group on G.In other words, a divisor is a formal sum v∈V (G) a v v, where a v ∈ Z for all v.If D is a divisor on G and if w ∈ V (G), then we use the notation D(w) to denote the coefficient a w of w in D. The support supp(D) of a divisor D is the set of all v for which D(v) = 0.
For two divisors D and D , we write The sets of all divisors and all effective divisors on G are denoted by Div(G) and Div + (G), respectively.
The degree of a divisor is the sum of its coefficients: deg(D Equivalence of divisors can also be described in terms of the following chip-firing game.An effective divisor D is interpreted as a distribution of chips over the vertices of G, where D(v) is the number of chips on v.
∈ A}| for all v ∈ A. If A is valid, then to fire A is to move chips from A to V (G) \ A, one for every edge of the cut (A, V (G) \ A).This yields a new divisor D given by D = D − L G 1 A , where 1 A is the characteristic vector of A. Since A is valid, this new divisor D is again effective.All equivalent effective divisors can be reached in this way: Lem. 2.3]).Let D, D be equivalent effective divisors.Then D can be obtained from D by subsequently firing an increasing sequence of valid subsets The rank of a divisor D ∈ Div(G) is defined as We have rank(D) = −1 if and only if D is not equivalent to an effective divisor.Given a graph G and an integer r ≥ 1, the r-th (divisorial) gonality dgon r (G) of G is the minimum degree of a rank r divisor on G.For r = 1, this is simply called the (divisorial) gonality of G: dgon(G) := dgon 1 (G).
1 Burn vertex q; 2 Burn all edges incident with burned vertices; 3 If a vertex v is incident with more burned edges than it has chips, burn v; 4 Repeat steps 2 and 3 until no more edges or vertices are burned; Algorithm 1: Dhar's burning algorithm for finite graphs.

Reduced divisors and Dhar's burning algorithm
Let G be a graph, and let q ∈ V (G).A divisor D ∈ Div(G) is called q-reduced if D(v) ≥ 0 for all v ∈ V (G) \ {q} and every non-empty valid set contains q.Every divisor D is equivalent to a unique q-reduced divisor; see [5,Prop. 3.1].A divisor D has rank at least 1 if and only if for every vertex v, the v-reduced divisor D v equivalent to D has at least one chip on v.
Dhar's burning algorithm [11], given in Algorithm 1, takes as input a graph G, a divisor D and a vertex q, and returns the maximal valid set A ⊆ V (G) \ {q}.In particular, the set A returned by Dhar's burning algorithm is empty if and only if D is q-reduced.

Metric graphs
A metric graph is a metric space Γ that can be obtained in the following way.Let G be a finite multigraph and let : E(G) → R >0 be an assignment of lengths to the edges of G. To construct Γ, take an interval [0, (e)] for every edge e ∈ E(G), and glue these together at the endpoints as prescribed by G. To turn it into a metric space, equip Γ with the shortest path metric in the obvious way.The metric graph Γ defined in this way will be denoted Γ(G, ).If = 1 is the unit length function, we write Γ(G) := Γ(G, 1).
If the metric graph Γ is constructed from the pair (G, ) as above, then we say that (G, ) is a model of Γ.We say that a model (G, ) is loopless (resp.simple) if G is loopless (resp.simple).The valency val(v) of v ∈ Γ is the number of edges incident with v in any loopless model (G, ) with v ∈ V (G).
A divisor on a metric graph Γ is an element of the free abelian group on Γ.In other words, a divisor is a formal sum v∈Γ a v v where a v ∈ Z for all v, and a v = 0 for all but finitely many v.The notations supp(D), deg(D), D ≥ D , Div(Γ), Div + (Γ) and Div d + (Γ) are defined analogously to the discrete case.
The definition of equivalence is a bit different.A rational function on Γ is a continuous piecewise linear function f : Γ → R with integral slopes.For each point v ∈ Γ, let a v be the sum of the outgoing slopes of f in all edges incident with v in some appropriate model of Γ.The corresponding divisor v∈Γ a v v is called a principal divisor.Two divisors D and D are equivalent if D − D is a principal divisor.
The rank of a divisor D ∈ Div(Γ) is defined as in the discrete case; that is: The r-th (divisorial) gonality dgon r (Γ) of Γ is the minimum degree of a rank r divisor on Γ.For r = 1, this is simply called the (divisorial) gonality of Γ: dgon(Γ) := dgon 1 (Γ).
If G is a finite graph and if Γ := Γ(G) is the corresponding metric graph with unit lengths, then two divisors D, D ∈ Div(G) are equivalent on G if and only if they are equivalent on Γ; see [3,Rmk. 1.3].Furthermore, in this case one has rank G (D) = rank Γ (D) for every divisor D ∈ Div(G); see [19,Thm. 1.3].

Rank-determining sets and strong separators
Let Γ be a metric graph, and let S ⊆ Γ be a subset.Following [20], we define the S-restricted rank of a divisor D ∈ Div(Γ) as where Div k + (S) is the set of degree k effective divisors whose support is contained in S. The set S is rank-determining if rank S (D) = rank(D) for all D ∈ Div(Γ).The following theorems are due to Luo.

Theorem 2.3 ([20, Thm. 1.10]). Let Γ, Γ be metric graphs, and let
We also formulate a discrete analogue of Theorem 2.2 for the case we have that C is a tree and for every s ∈ S there is at most one edge (in G) between C and s.Theorem 2.4 ([12,Lem. 2.6]).Let G be a graph, and let S ⊆ V (G) be a strong separator.
The following corollary is immediate from either Theorem 2.2 or Theorem 2.4.

Corollary 2.5. Let G be a loopless multigraph, and let H be a subdivision of
Fig. 1.A metric graph Γ = Γ(G, ) and a rescaling Γ = Γ(G, ) with = such that Γ and Γ are isometric.
3. Equivalence of the two forms of Conjecture 1.2 In this section, we prove Theorem 1.3 using a modification of the proof of [12,Thm. 5.1].The main idea is the following: given a rank r divisor D on the metric graph Γ(G), we will change the lengths of the edges between points in V (G) ∪ supp(D) in such a way that supp(D) is moved to the Q-points of the graph, all the while leaving the rank of D and the distances between the vertices of G unchanged.We will now make this precise.
We point out that Γ and its G-rescaling Γ can be isometric even if = .This is because vertices of degree 2 can be moved around, as illustrated in Fig. 1.In that case the vertex set V (G) is embedded into Γ ∼ = Γ in two different ways, and the divisor D and its G-rescaling D could be different divisors on the same metric graph.This will be the main tool in our proof of Theorem 1.3.
To rescale from real to rational edge lengths we use the following lemma.
Proof.Since the system has a solution x ∈ R >0 , the solution space {z | Az = b} is a non-empty affine Q-subspace of R n .Choose an affine rational basis y 0 , . . ., y d ∈ Q n for the solution space and write For every i ∈ {1, . . ., d}, choose a rational sequence {α for all k ≥ K 0 .This gives a sequence of solutions in Q n >0 converging to x.
We now come to the main result of this section, which is an extension of [12, Thm.
, then the length vector in (a) can be chosen in such a way that Γ is isometric to Γ.
Proof.(a) Write r := rank Γ (D), and let S ⊆ Γ be a finite rank-determining set.For every E ∈ Div r + (S), choose a divisor D E ∈ Div(Γ) and a rational function For E ∈ Div r + (S) and e ∈ E(G), let φ(f E , e) ∈ Z denote the slope of f E on e, in the forward direction of e.Note that a G-rescaling Γ(G, ) of Γ admits rational functions f E whose slope on e equals φ(f E , e), for all E ∈ Div r + (S) and all e ∈ E(G), if and only if y = is a solution to following system of equations: Since the coefficients (that is, φ(f E , e)χ C (e)) and constants (that is, 0) of this linear system are integral, and since is a solution, it follows from Lemma 3.2 that there exists a solution ∈ Q Consider the G-rescaling Γ := Γ(G, ).Let D be the corresponding G-rescaling of D, and let D E be the G-rescaling of D E for all E ∈ Div r + (S).By the above, we may choose rational functions f E on Γ such that the slope of f E on e equals φ(f E , e), for all E ∈ Div r + (S) and all e ∈ E(G).Then clearly (b) Choose a rational model ( G, ˜ ) of Γ.We repeat the argument of (a) with the following modifications.First, we add the requirement that V ( G) ⊆ V (G).Then every edge ẽ in G corresponds to a path in G, which we denote P ẽ.Second, we extend the linear system from (3.4) by adding the following constraints: y(e) = ˜ (ẽ), for all ẽ ∈ E( G). (3.5) Again, the coefficients and constants of the linear system are rational, and is a solution, so it follows from Lemma 3.2 that there is a solution ∈ Q >0 .The rest of the proof of (a) carries through unchanged, and the extra constraints from (3.5) ensure that Γ is isometric to Γ.
Proof of Theorem 1.3.Every rank r divisor on σ k (G) also defines a rank r divisor on Conversely, let D ∈ Div + (Γ(G)) be an effective divisor of rank r.By Theorem 3.
Remark 3.6.Analogously to the proof of the main result of [8], the linear system from the proof of Theorem 3.3(b) forms a certificate that dgon r (Γ) ≤ d.If r and d are fixed, then this certificate has size polynomial in the size of Γ, so it follows that Metric Divisorial r-Gonality for Q-graphs belongs to the complexity class NP. (For details, refer to the proof in [8].)Moreover, it can be deduced from the proof of [18,Thm. 3.5] that this problem is also NP-hard for r = 1 (see also [16,Thm. 1.3], where a different proof is given).We suspect that the same holds for all r ≥ 1.
Remark 3.7.The proof of Theorem 3.3(b) can also be used to find an upper bound on the size of the subdivision needed to get equality in Theorem 1.3.One such upper bound can be obtained by following the proof of [8,Cor. 6.2].We sketch a way to improve this bound.Let G be a graph with n vertices and m edges, let Γ := Γ( G) be the corresponding unit metric graph, and let D ∈ Div(Γ) be a divisor of degree d and rank r.We repeat the proof of Theorem 3.3(b) with respect to the rational model ( G, 1) and the rank-determining set S := V ( G) (use Theorem 2.2).Without loss of generality, we may assume that D is equal to one of the D E .Then the number of variables of the linear system is |E(G)| ≤ m + dn r .
Note that we can also allow a solution ≥ 0 instead of > 0. This has the effect of contracting some of the edges of the model G from the proof of Theorem 3.3, but the equations from (3.5) ensure that the resulting graph Γ is still isometric to Γ. Hence (3.4) and (3.5) determine a linear program Ax = b, x ≥ 0, and the entries of A are integers which can be shown to be bounded in absolute value by d.The set of feasible solutions is non-empty and bounded by (3.5), so there is a basic feasible solution x (see e.g.[21,Thm. 4.2.3]).Hence there is a subset B ⊆ {1, . . ., |E(G)|} such that x B = A −1 B b and x B c = 0. Therefore the lowest common denominator of the entries of x is at most In conclusion, if the unit metric graph Γ = Γ( G, 1) has a divisor of rank r and degree d, then so does σ k ( G) for some k ≤ (m + dn r )! • d m+dn r .

A graph G such that dgon(σ 2 (G)) < dgon(G)
In this section we construct a class of graphs, which we call "tricycle graphs".We show that the divisorial gonality of any tricycle graph G is strictly greater than the divisorial gonality of its 2-subdivision σ 2 (G), and thus of its associated metric graph Γ(G).
• Subdivide the six edges incident with v 0 .In what follows, we will show that every tricycle graph G satisfies dgon(G) = 6 and dgon(Γ(G)) = dgon(σ 2 (G)) = 5.First of all, we exhibit a positive rank divisor of degree 5 on σ 2 (G).Proof.Let D 0 ∈ Div(σ 2 (G)) be the effective divisor with two chips on v 0 and one chip on the midpoint of each of the transition edges v

We call the vertices v
Then deg(D 0 ) = 5.In light of Corollary 2.5, in order to show that D 0 has positive rank, it suffices to prove that D 0 reaches v 0 and the transition vertices (c) the minimal simple tricycle T ms .
Fig. 2. A generic tricycle, the minimal tricycle, and the minimal simple tricycle, with the transition vertices and the transition edges highlighted for emphasis.
Clearly D 0 reaches v 0 , for we have D 0 (v 0 ) > 0. Now fix i ∈ {1, 2, 3}.To reach the transition vertices v − i and v + i , let S i ⊆ V (σ 2 (G)) be the connected component of σ 2 (G) \ supp(D 0 ) that contains the cycle C i .Then the subset S c i can be fired, and doing so yields an effective divisor . This shows that D 0 reaches v − i and v + i , for all i ∈ {1, 2, 3}.It follows that D 0 is a positive rank divisor on σ 2 (G), hence dgon(σ 2 (G)) ≤ 5.
Evidently, the divisor from Proposition 4.2 is not supported on vertices of G.The remainder of this section is dedicated to showing that G has no positive rank divisors of degree 5. Along the way, we also prove that dgon(Γ(G)) ≥ 5.
In Lemma 4.5 below, we show that every positive rank v 0 -reduced divisor of degree at most 5 on a subdivision of the minimal tricycle T m must be of a very specific form.This will subsequently be used to show that dgon(Γ(G)) = dgon(σ 2 (G)) = 5 (see Corollary 4.7) and dgon(G) = 6 (see Theorem 4.8) for every tricycle graph G.
For convenience, we use the following notation.The following simple lemma is essential to our proof, and will be used repeatedly.Lemma 4.4.Let G be a graph and let v 0 ∈ V (G).Let e 1 , . . ., e k ∈ V (G) be the edges incident with v 0 , and let v i ∈ V (G) \ {v 0 } be the other endpoint of e i for every i.
Moreover, let H be a subdivision of G, let D ∈ Div(H) be a positive rank v 0 -reduced divisor on H, and let w ∈ V (H) be a vertex with D(w) = 0. Then an execution of Dhar's burning algorithm on the triple (H, D, w) has the following properties: Proof.(a) Since D has positive rank and D(w) = 0, the divisor D cannot be w-reduced, so Dhar's algorithm returns a non-empty subset A ⊆ V (G) that can be fired.Since D is v 0 -reduced, we must have v 0 ∈ A, which means that v 0 is not burned.(b) Partition I as I = I 0 ∪ I 1 , where i ∈ I 0 if all vertices of the path P e i (v 0 ,v i ] are burned, and i ∈ I 1 otherwise.Since v 0 is not burned, it has at most D(v 0 ) burning neighbours, so |I 0 | ≤ D(v 0 ).Moreover, if i ∈ I 1 , then v i is burned, but not all vertices of the path P e i (v 0 ,v i ] are burned, so there must be at least one chip on P e i (v 0 ,v i ) .The conclusion follows.
We will apply Lemma 4.4 to an arbitrary subdivision of the minimal tricycle T m .For this we use the following terminology.Using notation from Definition 4.3, if H is a subdivision of T m , then the three transition edges in H, which we call the transition paths.The transition vertices of H are the images in H of the original six transition vertices Proof.First, we prove that there must be at least one chip on every transition path.Suppose, for the sake of contradiction, that one of the transition paths, say , has no chips at all.We start an execution of Dhar's burning algorithm on (H, D, v + 1 ).Let H + 2 ⊆ H be the union of the cycle C 2 and the transition path . We claim that the number of chips on H + 2 plus the number of burned transition vertices in H + 2 is at least 3. To that end, note first of all that v − 2 is burned, since there is no chip on the transition path . Now we distinguish three cases: 2 is not burned, then there must be at least two chips on C 2 to stop the fire spreading from v − 2 to v + 2 .In this case, H + 2 contains at least one burned transition vertex (namely v − 2 ) and at least two chips, for a total of at least 3.
• If v + 2 is burned but v − 3 is not burned, then there must be at least one chip on the half-open transition path . In this case, H + 2 contains two burned transition vertices (v − 2 and v + 2 ) and at least one chip, for a total of at least 3; • If both v + 2 and v − 3 are burned, then H + 2 contains three burned transition vertices (v − 2 , v + 2 and v − 3 ).
Likewise, write . Analogously, the number of chips plus the number of burned transition vertices on H − 1 is at least 3. Since H + 2 and H − 1 are disjoint, the total number of chips on the outer ring plus the total number of burned transition vertices is at least 6.But since the transition vertices are exactly the T m -neighbours of v 0 , and since the half-open paths P [v 0 ,v ± i ) are disjoint from the outer ring, it follows from Lemma 4.4(b) that deg(D) ≥ 6, which is a contradiction.We conclude that every transition path must have at least one chip.
Second, we prove that there must be two chips on v 0 .Since the total number of chips is at most 5, there must be a cycle C i on the outer ring with at most one chip.Choose w ∈ V (C i ) with D(w) = 0 and start an execution of Dhar's burning algorithm on (H, D, w).Since there is at most one chip on C i , the entire cycle C i is burned.It follows from Lemma 4.4(b) that there are at least two chips on . Therefore the number of chips on the outer ring is at most 3, so there must be another cycle C j (j = i) on the outer ring with at most one chip.By an analogous argument, there are at least two chips on . But since the outer ring has at least 3 chips (one on every transition path), there can be at most 2 chips on . The only way to meet these requirements is if there are exactly two chips on v 0 .
To conclude the proof, note that 2 chips on v 0 and at least 1 chip on every transition path add up to at least 5 chips in total.Since deg(D) ≤ 5, all chips have been accounted for.In particular, there cannot be more than one chip on each of the transition paths.All that remains is to prove that every tricycle graph has divisorial gonality 6.To do so, we once again use the preceding lemmas.Proof.Suppose, for the sake of contradiction, that dgon(G) ≤ 5. Then we may choose a positive rank v 0 -reduced divisor D ∈ Div(G) with deg(D) ≤ 5. We interpret G as a subdivision of the minimal tricycle T m .It follows from Lemma 4.5 that D has two chips on v 0 and exactly one chip on every transition path.Since G is a tricycle graph, the transition edges of T m are not subdivided.Therefore a chip on a transition edge must lie on one of the transition vertices.
By the above, the divisor D has between 0 and 2 chips on each of the cycles C 1 , C 2 , C 3 on the outer ring, and all such chips must lie on the transition vertices.Since the total number of chips on the outer ring is odd, there must be a cycle C i with exactly one chip.Assume without loss of generality that C 1 is such a cycle, and that D(v − 1 ) = 0 and D(v + 1 ) = 1.We start an execution of Dhar's burning algorithm on (G, D, v − 1 ).Since there is only one chip on C 1 , the entire cycle C 1 is burned.In particular, the vertex v + 1 is burned.The transition edge v + 1 v − 2 has exactly one chip, which is on v + 1 , so the fire spreads via this edge to the vertex v − 2 , which is also burned.But now we see that at least three T mneighbours of v 0 are burned (namely, v − 1 , v + 1 and v − 2 ), so it follows from Lemma 4.4(b) that there must be at least 3 chips on . This is a contradiction, and we conclude that dgon(G) ≥ 6.
To see that dgon(G) ≤ 6, note that the set {v − 3 } of all transition vertices is a strong separator.Therefore the effective divisor with one chip on each of the transition vertices has positive rank, by Theorem 2.4, so dgon(G) ≤ 6.

A family of examples with larger gaps
In this section, we combine tricycle graphs in a certain way in order to obtain graphs G k with dgon(G k ) = 6k and dgon(σ 2 (G k )) = dgon(Γ(G k )) = 5k, which shows that the gap between dgon(Γ(G)) and dgon(G) can be arbitrarily large.Furthermore, we show that dgon r (Γ(G)) and dgon r (G) differ by at most a factor 2. Definition 5.1.Given a (connected) simple graph H and an integer t ≥ 1, an (H, t)skewered graph is a graph G that can be obtained in the following way: • Start with a disjoint union of graphs G 1 , . . ., G n , where n = |V (H)|.
• For every i ∈ [n], choose a base vertex w i ∈ V (G i ); • For every edge ij ∈ E(H), add t parallel edges between w i and w j , and subdivide these edges in an arbitrary way.
An example of a (K 2 , 12)-skewered graph is given in Fig. 3 Proof.First, we prove that dgon(G) ≤ dgon(G i ).We prove that D has positive rank.By Corollary 2.5, it suffices to prove that D reaches all vertices of every G i .Let v ∈ V (G i ), and choose an effective divisor D i ∈ Div(G i ) equivalent to D i with D i (v) > 0. By Proposition 2.1, we can go from D i to D i by subsequently firing an increasing sequence Then, starting with D and subsequently firing the sets In other words, we can play the chipfiring game on G i while leaving the remainder of G unchanged.This shows that D reaches all vertices of every G i , so it follows from Corollary 2.5 that rank(D) ≥ 1.

|V (H)| i=1
dgon(G i ).Suppose, for the sake of contradiction, that D ∈ Div(G) is a positive rank w 1 -reduced divisor with deg(D) <

|V (H)| i=1
dgon(G i ).We claim that D is w i -reduced for all i.To that end, let S ⊆ V (G) be a subset for which there is some ij ∈ E(H) with w i ∈ S and w j / ∈ S. Since there are t parallel paths in G between w i and w j , it follows from the max-flow min-cut theorem that , so S cannot be fired.Thus, if S ⊆ V (G) is a subset which can be fired, then w 1 ∈ S (because D is w 1 -reduced), and therefore w i ∈ S for all i (because H is connected).This proves our claim that D is w i -reduced for all i.
Next, we claim that D restricts to a positive rank divisor on every G i .Indeed, let v ∈ V (G i ) for some i, and choose an equivalent effective divisor D ∈ Div(G) with D (v) > 0. By Proposition 2.1, we can go from D to D by subsequently firing an increasing sequence U 1 ⊆ • • • ⊆ U k of valid sets.Since D is w i -reduced, we have w i ∈ U 1 , and therefore w i ∈ U j for all j.Since w i is the only vertex in G i connected to anything outside of G i , the firing sequence U 1 ⊆ • • • ⊆ U k only ever sends chips out of G i , and never into G i .Hence it restricts to a valid firing sequence in G i , which shows that the restricted divisor D| G i ∈ Div(G i ) reaches v.This proves our claim that D restricts to a positive rank divisor on every G i .But now it follows that deg(D) ≥

|V (H)| i=1
dgon(G i ), contrary to our assumption.This is a contradiction.
Proof of Theorem 1.4.Let G 1 , . . ., G k be tricycle graphs, and let H be an arbitrary connected simple graph on k vertices.Choose t ≥ 6k, and let G be an (H, t)-skewered graph obtained from the graphs G 1 , . . ., G k .Then it follows from Lemma 5.2 that dgon(G) = 6k.Furthermore, for every s ∈ N 1 , the subdivided graph σ s (G) is an (H, t)-skewered graph relative to the base graphs σ s (G 1 ), . . ., σ s (G k ), so it follows from Lemma 5.2 and Corollary 4.7 that A simple and bipartite realization can be obtained by choosing the tricycles G 1 , . . ., G k simple and bipartite (e.g. the tricycles skewered together in Fig. 3), and choosing an appropriate subdivision in the process of Definition 5.1.Theorem 1.4 shows that the discrete and metric divisorial gonality can be arbitrarily far apart.The following simple result shows that large gaps like this can only occur when the metric gonality is also large.The small blue hexagons represent the chips of an optimal divisor on the 2-regular subdivision.In each example, the divisorial gonality of the original graph is one higher.(For interpretation of the colours in the figure(s), the reader is referred to the web version of this article.)

Computational results and open questions
Apart from the tricycle graphs, we have found a few other counterexamples, which we sketch here.First of all, the proofs from §4 still hold if each of the cycles C 1 , C 2 and C 3 is replaced by any graph C which has two distinct vertices v − , v + such that: (i) there are two edge-disjoint paths between v − and v + ; (ii) the divisor v − + v + has positive rank on C.
Second, we have found a number of counterexamples which we have verified computationally, but for which we have no formal proof.Most of these have a structure very similar to a tricycle graph: there are 3 cycles which are connected to one another and to a central vertex in some way.A small selection of these counterexamples is given in Fig. 4. In each of these, the optimal divisor on the 2-regular subdivision σ 2 (G) has 3 chips on the midpoints of certain edges, and 2 or 3 chips on the central vertex, and dgon(G) = dgon(σ 2 (G)) + 1.Note that the counterexample depicted in Fig. 4(c) is 3regular.We have also found counterexamples where the outer ring has 5 or 7 cycles; see Fig. 4(d).We have not found a counterexample with 9 or more cycles on the outer ring.See [13] for code and additional figures.
We have tested Conjecture 1.2(a) for k = 2 and r = 1 for all simple connected graphs on at most 10 vertices.These graphs were generated using the program geng from the gtools suite packaged with nauty [22,23], and tested using custom code that we wrote to compute the divisorial gonality of a graph [13].We have found that every simple connected graph with 9 or fewer vertices satisfies dgon(σ 2 (G)) = dgon(G), and that there are exactly 29 counterexamples with 10 vertices (and no parallel edges), including the minimal simple tricycle T ms and the graphs depicted in Fig. 4(e)-(h).For a list of all 29 minimal simple counterexamples and code to reproduce this list, see [13].There we have also included optimized code to check whether the divisorial gonality of a given graph satisfies the Brill-Noether bound, which we have used to verify Conjecture 1.1 for all simple connected graphs with at most 13 vertices.No counterexamples were found.
We close with a few open problems.
(a) There exists a loopless model (G, ) with supp(D) ⊆ V (G) and a rational length vector contains all points of non-linearity of f E , for every E ∈ Div r + (S).Choose an orientation of the edges of G.For every cycle C in G, choose a circular orientation of the edges of C, and define χ C : E(G) → {−1, 0, 1} by setting χ C (e) := ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 1, if e ∈ E(C) and the orientations of G and C agree on e; −1, if e ∈ E(C) and the orientations of G and C disagree on e; 0, if e / ∈ E(C).
(e) y(e) = 0, for every cycle C and every E ∈ Div r + (S).(3.4) 3(b), there exists a divisor D ∈ Div + (Γ(G)) with deg(D ) = deg(D) and rank(D ) ≥ rank(D) which is supported on the Q-points of Γ(G).Then D is supported on the vertices of the model (σ k

Definition 4 . 1 .
A tricycle graph is a multigraph G that can be obtained in the following way:• Start with three disjoint cycles C 1 , C 2 , C 3 , each on at least 2 vertices (a cycle on 2 vertices consists of two vertices connected by two parallel edges).• Choose two distinct vertices on each of these cycles, say v transition edges, and v 0 the central vertex.The outer ring is the union of the cycles C 1 , C 2 , C 3 and the transition edges.

Fig. 2
Fig. 2 illustrates an example of a tricycle graph, along with the minimal tricycle T m and the minimal simple tricycle T ms .Note that a multigraph (resp.simple graph) G is a tricycle if and only if G can be obtained by taking a subdivision of the minimal tricycle T m (resp.the minimal simple tricycle T ms ) in such a way that the transition edges are not subdivided.In what follows, we will show that every tricycle graph G satisfies dgon(G) = 6 and dgon(Γ(G)) = dgon(σ 2 (G)) = 5.First of all, we exhibit a positive rank divisor of degree 5 on σ 2 (G).

Definition 4 . 3 .
Let G be a graph and let H be a subdivision of G.For e = uw ∈ E(G), let P e [u,w] ⊆ H denote the path uv 1 v 2 • • • v k w inH corresponding to the subdivided edge e.Furthermore, let P e (u,w) := P e [u,w] \{u, w}, P e [u,w) := P e [u,w] \{w} and P e (u,w] := P e [u,w] \{u} denote the corresponding open and half-open subpaths.If e is the only edge between u and w, then we omit the superscript and simply write P [u,w] , P (u,w) , P [u,w) and P (u,w] . or in other words, the endpoints of the transition paths in H. (This is consistent with our definition of the transition vertices of a tricycle graph, which can also be seen as a subdivision of T m .)Lemma 4.5.Let H be a subdivision of the minimal tricycle T m .If D ∈ Div(H) is a positive rank v 0 -reduced divisor with deg(D) ≤ 5, then D must have two chips on v 0 and exactly one chip on each of the transition paths
|V (H)| i=1 dgon(G i ).For every i, choose a positive rank divisor D i ∈ Div(G i ) of minimum degree.This defines a divisor D ∈ Div(G) with deg(D) = |V (H)| i=1

Proposition 5 . 3 .
Let G be a graph.For every r ≥ 1, one has dgon r (G) ≤ 2 dgon r (Γ(G)) − r.Proof.Let D 1 ∈ Div(Γ(G)) be a divisor of rank r and degree d := dgon r (Γ(G)).Choose some E ∈ Div r + (G), and choose a divisorD 1 ∼ D 1 such that D 1 ≥ E. Let D 2 ∈ Div(G)be the divisor obtained from D 1 by replacing every chip on the interior of some edge uv ∈ E(G) by one chip on u and one chip on v. Since D 1 ≥ E and supp(E) ⊆ V (G), the divisor D 1 has at least r chips on vertices of G, so deg(D 2 ) ≤ 2d − r.By firing everything but the interior of the edge uv, we can move the newly added chips on u and v so that one of the two reaches the original position of the chip in D 1 and the other becomes superfluous.This shows that D 2 is equivalent on Γ to a divisor D 2 with D 2 ≥ D 1 , so rank G (D 2 ) = rank Γ (D 2 ) ≥ r, by [19, Thm.1.3].

Fig. 4 .
Fig.4.Additional counterexamples to Conjecture 1.2(a) for k = 2 and r = 1.The small blue hexagons represent the chips of an optimal divisor on the 2-regular subdivision.In each example, the divisorial gonality of the original graph is one higher.(For interpretation of the colours in the figure(s), the reader is referred to the web version of this article.) below.Lemma 5.2.Let G be an (H, t)-skewered graph with t ≥