Graphs of gonality three

In 2013, Chan classified all metric hyperelliptic graphs, proving that divisorial gonality and geometric gonality are equivalent in the hyperelliptic case. We show that such a classification extends to combinatorial graphs of divisorial gonality three, under certain edge- and vertex-connectivity assumptions. We also give a construction for graphs of divisorial gonality three, and provide conditions for determining when a graph is not of divisorial gonality three.

Our main result in this paper is an analogue of Theorem 1.1 for graphs of divisorial gonality 3. Although Theorem 1.1 is stated for metric graphs, ours holds only for combinatorial graphs, without the data of lengths associated to the edges. For one portion we need an additional assumption on our graph G called the zero-three condition, defined at the beginning of Section 4. Theorem 1.2. Consider the following 3 conditions: 1. G has (divisorial) gonality 3.
2. There exists a non-degenerate harmonic morphism ϕ : G → T where deg(ϕ) = 3 and T is a tree.
3. There exists a cyclic automorphism σ : G → G of order 3 that does not fix any edge of G satisfying the property that G/σ is a tree.
The decision to restrict our attention to 3-vertex-connected graphs in the simple case is in part supported by Proposition 4.5, which shows that a simple, bridgeless, trivalent graph that is not 3-vertex-connected must have gonality at least 4. Moreover, the example graph in Figure 4.1 shows that 3-edge-connectedness is not a strong enough assumption to guarantee the existence of a cyclic automorphism of order 3; and the example graph in Figure 4.2 shows that the zero-three condition is necessary for (1) implies (3), even for 3-vertexconnected graphs. To justify our 3-edge-connected assumption for the multigraph case, we point to recent work by Corry and Steiner, appearing in [13,Theorem 10.24], which shows that for d-edge-connected graphs with more than d vertices, the set of degree d non-degenerate harmonic morphisms to a tree is in bijection with divisors of degree d and rank 1 on the graph. With some extra work to rule out the possibility of hyperellipticity, this result can be used to prove our main theorem for multigraphs. However, the proof we present is independently formulated.
Our paper is organized as follows. In Section 2, we establish definitions and notation, and review previous results on divisors and harmonic morphisms of graphs. In Section 3, we prove Theorem 3.1, which is the first part of Theorem 1.2 and applies to general multigraphs. In Section 4, we restrict our attention to simple graphs in order to prove Theorem 4.1, which adds the third condition in Theorem 1.2. We also give a criterion for identifying graphs with gonality strictly greater than 3. Finally, in Section 5, we present a construction for a (proper) subset of graphs of gonality 3.

Definitions and Notation
We define a graph G = (V, E) with vertex set V (G) and edge set E(G) to be a finite, connected, loopless, multigraph. Throughout this paper, all graphs are assumed to be combinatorial (that is, without lengths assigned to edges) unless otherwise stated. Graphs with no multiedges are called simple. Given a vertex v ∈ V (G) and an edge e ∈ E(G), we use the notation v ∈ e to indicate that v is an endpoint of e. A graph G = (V, E) is k-edge-connected if, for any set W of k − 1 edges, the subgraph (V, E − W ) is connected. We let η(G) denote the edge-connectivity of the graph. That is, η(G) is the maximum integer k such that G is k-edge-connected. A bridge of G is an edge whose deletion strictly increases the number of connected components of G. A graph is bridgeless if it has no bridges, or equivalently if it is 2-edge-connected.
Similarly, a graph G = (V, E) is k-vertex-connected (or just k-connected ) if, for any set U of k −1 vertices, the subgraph (V − U, E) is connected. We let κ(G) denote the vertex-connectivity of the graph. That is, κ(G) is the maximum integer k such that G is k-vertex-connected. (By convention, we set κ(G) = |V | − 1 if every pair of vertices in G is joined by an edge.) Since removing a vertex from a graph removes all edges incident to that vertex, we have that κ(G) ≤ η(G) for any graph G.

Divisor Theory on Graphs
We now review the key concepts of divisor theory on graphs, as developed in [1]. A divisor D on a graph G is a Z-linear combination of vertices. We will often explicitly write out divisors with the notation where D(v) denotes the value of D at v. The set of all divisors Div(G) on a graph G forms an abelian group under component-wise addition. The degree of a divisor D is defined as the sum of its integer coefficients: For a fixed k ∈ Z, let Div k (G) be the set of all divisors of degree k on G. A divisor D is effective if, for all v ∈ V (G), D(v) ≥ 0. Let Div + (G) be the set of all effective divisors on a graph G and for k ∈ Z >0 , let Div k + (G) be the set of all effective divisors of degree k on G. For a given effective divisor D, we define the support of D as The Laplacian L(G) of a graph G is the |V | × |V | matrix with entries We use ∆ : Div(G) → Div(G) to denote the Laplace operator associated with the Laplacian matrix. A principal divisor is a divisor in the image of ∆. We use Prin(G) to denote the set of principal divisors on a graph G, i.e. Prin(G) = ∆(Div(G)). Notice that Prin(G) is a normal subgroup of Div 0 (G). We can therefore define the Jacobian Jac(G) of a graph G as the quotient group Div 0 (G)/Prin(G). Now, define an equivalence relation ∼ on divisors such that D ∼ E if and only if D − E ∈ Prin(G). We say in this case that D and E are linearly equivalent and define the linear system associated with a divisor D as For a divisor D ∈ Div(G), we define the rank of D as r(D) := −1 if |D| = ∅, and otherwise as The gonality of a graph G is defined as gon(G) := min{deg(D) : D ∈ Div + (G), r(D) ≥ 1}.
Later, when we need to distinguish between two different types of gonality, this will be referred to as divisorial gonality.

Baker-Norine Chip-Firing
The definition of gonality provided in the previous section has an equivalent statement in terms of chip-firing games on graphs. In a chip-firing game, we think about placing integer numbers of poker chips on the vertices of our graph. A negative number of poker chips corresponds to a vertex being "in debt". A chip-firing move involves selecting a vertex v ∈ V (G), subtracting val(v) chips from v, and adding |E(v, v )| chips to each v adjacent to v. The Baker-Norine chip-firing game is played with the following rules: 1. One player places k chips on the vertices V (G) of a graph G in any arrangement.
2. Another player (the adversary) chooses a vertex v ∈ V (G) from which to subtract a chip, possibly putting v into debt.
3. The first player wins if they can reach a configuration of chips where no vertex is in debt via a sequence of chip-firing moves. Otherwise, the adversary wins.
Notice that these "chip configurations" correspond to divisors on graphs. By standard results as in [2], chip-firing moves correspond to subtracting principal divisors; the divisors present before and after chipfiring are equivalent; and the gonality of a graph is equivalent to the minimum number of chips k required to guarantee that the first player has a winning strategy in the Baker-Norine chip-firing game. Hence, we define a winning divisor D to be a divisor satisfying r(D) ≥ 1.
Since chip-firing is a commutative operation, we can chip-fire from an entire subset A ⊂ V (G) at once by sending a chip along each edge outgoing from the subset. Let 1 A denote the indicator vector on A.
Then, given a divisor D, the resulting divisor after chip-firing from the subset A is D − ∆1 A . We define the outdegree of A from a vertex v ∈ A to be the number of edges leaving A from v, so Hence, a chip-firing move from a subset The following result is proven in [23]. This means that if we have a divisor D with r(D) ≥ 1, then we can move at least one chip onto every vertex of our graph (in turn) without ever putting any of the vertices of the graph into debt. For a given divisor D, we say D is v-reduced with respect to some vertex v ∈ V (G) if This means that every vertex (except possibly v) is out of debt, and that there exists no way to fire from any subset of V (G) − {v} without inducing debt. The following two results are proven in [2]: Lemma 2.2. Given a divisor D ∈ Div(G) and a vertex v ∈ V (G), there exists a unique v-reduced divisor D such that D ∼ D.
We will use Red v (D) to denote this unique v-reduced divisor. Thus, we can determine if a divisor is winning divisor by checking that, for each v ∈ V (G), the associated v-reduced divisor satisfies v ∈ supp(Red v (D)). Furthermore, given a divisor D and a vertex v for which D is effective away from v, Algorithm 1, developed by Dhar in [14], computes Red v (D).
We offer the following intuitive explanation of Algorithm 1. We begin with a graph G, a vertex v, and a divisor D, which is assumed to be effective away from v. Then we "start a fire" at the vertex v. As the fire spreads through the graph, chips on vertices act as "firefighters", protecting their vertex from the encroaching flames. To determine which vertices and edges of the graph catch on fire, we repeat the following two steps until no new vertices or edges burn.
1. If an edge is adjacent to a burning vertex, then that edge also catches fire and begins to burn.
2. If a vertex is adjacent to more burning edges than it has chips, then that vertex begins to burn.
Once a stable state is reached, we chip-fire from the set of unburnt vertices. Then we begin the burning process again starting at v. If at any point the entire graph burns, the algorithm terminates and outputs the resulting divisor.
We refer the reader to [4] for a proof that Algorithm 1 terminates and that the resulting divisor is indeed Red v (D). As a corollary of Lemma 2.3, we have the following result.
Corollary 2.4. For an effective divisor D ∈ Div + (G), if there exists some v ∈ V (G) such that v / ∈ supp(D) and for which beginning Dhar's burning algorithm at v results in the entire graph burning, then r(D) < 1.

Riemann-Roch for Graphs
For a graph G, we define the canonical divisor as The canonical divisor has degree 2g(G) − 2. In [2], Baker and Norine prove the following Riemann-Roch theorem for graphs, analogous to the classical Riemann-Roch theorem on algebraic curves: Notice that this implies r(K) = g(G) − 1. As a consequence, we can prove the following result: Proposition 2.6. If G is a graph with genus g(G) ≤ 2, then gon(G) ≤ 2.
Proof. If g(G) = 0, then G must be a tree, giving gon(G) = 1. If g(G) = 1 and D ∈ Div(G) satisfies deg(D) = 2, then by Riemann-Roch for graphs, we see that

Harmonic Morphisms of Graphs
We now turn to another notion of gonality called geometric gonality, which is defined in terms of maps between graphs. If G and G are graphs, a morphism ϕ : , satisfying the following two conditions: This definition comes from [3]. Morphisms defined on graphs are sometimes indexed, as in [12]. In this paper, we will only consider non-indexed morphisms. For a vertex v ∈ V (G), we define the multiplicity of ϕ at v with respect to an edge e ϕ(v) as We define the degree of a harmonic morphism to be We remark that there are multiple inequivalent notions of geometric gonality defined in the literature. In particular, some authors consider refinements of the original graph [10], while other authors only consider graph morphisms that are also homomorphisms [22, Section 1.3]. The results in our paper hold specifically for the definition of geometric gonality given above.

Bounds on Gonality
The following result is stated in [12] and proven here for the reader's convenience.
Proof. Suppose that D ∈ Div + (G) is a divisor with deg(D) < min{|V (G)|, η(G)}. This means that D does not contain all of the vertices of G in its support, nor can we fire from any subset of supp(D) because any such subset A ⊆ supp(D) will have outdeg A (A) > v∈A D(v). Hence, D is not a winning divisor.
The treewidth tw(G) of a graph G is defined to be the minimum width amongst all possible tree decompositions of G. The following result is proven in [24].
It is shown in [7] that, for a simple graph G, tw(G) ≥ min{val(v) : v ∈ V (G)}. Hence, we have the following result.
We also have the following "trivial" upper bound on gonality.
This upper bound is typically only attained when the edge-connectivity of the graph is high relative to the number of vertices. In fact, if G has a vertex v which is not incident to any multiple edges, then gon(G) ≤ |V (G)| − 1, since placing one chip on every vertex except v results in a winning divisor.

Multigraphs of Gonality Three
In this section, we will prove the following result, which is simply the first part of Theorem 1.2 and applies to all multigraphs of edge-connectivity at least 3.
Theorem 3.1. If G is a 3-edge-connected graph, then the following are equivalent: 1. G has gonality 3.
One approach to proving this result would be to apply [13, Theorem 10.24] with d = 3 for the direction (1) implies (2), and then argue that if (2) holds then hyperellipticity is not possible. Here we present a different approach, which will also lay the groundwork for the subsequent section. We will first prove some preliminary results, which will allow us to define an equivalence relation on the vertices of G. From here, the map from G to the resulting quotient graph provides our non-degenerate harmonic morphism.
Proof. Note that (1) comes as a corollary of Proposition 2.6. For (2), assume that |V (G)| ≥ 4 and η(G) ≥ 4. Then, by Lemma 2.7, we have gon(G) ≥ 4. If |V (G)| < 3, we know that G is either a single point or the path P 2 (both of which have gonality 1), or that G is a banana graph on two vertices, which has gonality 2 (see As an aid for the reader, we introduce the 3-edge-connected graph depicted in Figure 3.2. After proving each of the following lemmas, we will demonstrate the effect of the result on this graph, culminating in the construction of a non-degenerate harmonic morphism down to a tree. For the next two lemmas, let G be a simple, 3-edge-connected graph with gon(G) = 3, and let D be a divisor on G of rank 1 and degree 3.
Proof. Since r(D) = 1, we know that for any vertex . Thus for any v ∈ V (G), there exists at least one divisor D ∈ Div + (G) such that D ∼ D and v ∈ supp(D ).
For uniqueness of D , consider the Abel-Jacobi map S (k) : Div k + (G) → Jac(G) with basepoint v 0 , defined as follows: Since G is 3-edge-connected, the Abel-Jacobi map with basepoint v 1 is injective, and so up to relabelling we Note that a generalization of Lemma 3.3 for divisors of degree d and rank 1 on d-edge-connected graphs for arbitrary d ∈ Z >0 is proven in [13].
The equivalence classes associated with this relation are We define a morphism ϕ : G → G/ ∼ D in the following way: In our running example from Figure 3.2, define D to be the divisor with one chip on every vertex in the left-most 3-cycle of the graph. Figure 3.3 shows the partitioning of vertices into equivalence classes on our running example graph, as well as the effect of ϕ on the graph.
Since we will use this quotient morphism in our proof of Theorem 3.1, we now prove the following lemma, which will aid us in showing that this morphism is harmonic.
Proof. Suppose we have an edge e = uv ∈ E(G) and we begin with the divisor D u satisfying supp(D u ) = [u] D . By Lemmas 2.1 and 2.2, there exists a unique v-reduced divisor equivalent to D u which can be reached by a finite sequence of chip-firing moves. Furthermore, by Lemma 3.21 of [23], we never need to fire from the vertex v itself during the reduction process. Since u and v are connected by an edge, our first chip-firing move must move at least one chip onto v; otherwise, we would have fired a collection of vertices not including u, thereby obtaining another effective divisor D with u ∈ supp(D ), which contradicts the uniqueness of D u . However, by the uniqueness of the divisor D v with v ∈ supp(D v ), we must have moved all three chips Note that in our running example, the partition of vertices depicted in Figure 3.3(a) shows that there are exactly three edges between every pair of adjacent vertex classes, and that the number of edges incident with each vertex v in the class is precisely the number of chips on v in the associated divisor D v .
Armed with these results, we can now prove the main result of this section.
Proof of Theorem 3.1. We will first show that (1) =⇒ (2). Let G be a graph of gonality 3. If |V (G)| ≤ 3, then by the proof of Lemma 3.2, we know that gon(G) = 3 only if |V (G)| = 3. Assume now that |V (G)| > 3. Since gon(G) = 3, there exists a divisor D ∈ Div + (G) such that deg(D) = 3 and r(D) = 1. Define the equivalence relation ∼ D as before, with [v] D again referring to the equivalence class associated to v under ∼ D . Let ϕ be the quotient morphism ϕ : G → G/ ∼ D defined above. We will now show that ϕ is a non-degenerate harmonic morphism of degree 3.
By Lemma 3.4, we have The assumption that |V (G)| > 3 ensures that we have at least one edge between vertices in different equivalence classes.
For the reverse direction (2) It is clear that D is effective and by Lemma 2.13 in [3], deg(D) = 3. We claim that r(D) ≥ 1. Pick x ∈ G.
Theorem 3.1 can be applied to determine the geometric gonalities of graphs with known divisorial gonalities. For example, consider the 3-cube graph Q 3 illustrated in Figure 3.4, which is 3-edge-connected. It can be computationally verified that gon(Q 3 ) = 4. Since this graph is not a tree and doesn't have divisorial gonality 2 or 3, we know by Theorems 1.1 and 3.1 that ggon(Q 3 ) ≥ 4.  We can also apply Theorem 3.1 to certain graphs with bridges, assuming that they become 3-edgeconnected after contracting these bridges. This is due to the following proposition, which comes as an immediate consequence of Corollary 5.10 in [3] on rank-preservation under bridge contraction:

Simple Graphs of Gonality Three
We now restrict our attention to graphs that are simple. We say a graph of gonality 3 satisfies the zero-three condition if there exists a divisor D such that deg(D) = 3, r(D) = 1, and for any three distinct vertices a, b, c with D ∼ (a) + (b) + (c), we have that a, b, c either share 0 edges or share 3 edges.
The following theorem extends Theorem 3.1 by adding an extra equivalent statement, under a few stronger assumptions.
Theorem 4.1. If G is a simple, 3-vertex-connected combinatorial graph satisfying the zero-three condition, then the following are equivalent: 1. G has gonality 3.
3. There exists a cyclic automorphism σ : G → G of order 3 that does not fix any edge of G, such that G/σ is a tree.
Notice that we no longer need to worry about the case where |V (G)| = 3; this is because there are no simple 3-vertex-connected graphs with exactly 3 vertices. Also note that while statements (1) and (2) in Theorem 4.1 are nearly identical to those given in Theorem 1.1, statement (3) now requires the extra condition that the automorphism σ does not fix any edge of G. In our proof of this theorem, we will show that this condition is required for the implication (3) =⇒ (2) to hold.  We should also remark why we need the stronger assumptions that our graph is 3-vertex-connected instead of 3-edge-connected, and that our graph satisfies the zero-three condition. Consider the graph G in Figure 4.1. The divisor (v 1 ) + 2(v 2 ) has positive rank, so G has gonality at most 3; and since G has K 4 as a minor, the treewidth of the graph, and thus its gonality, is at least 3 by [6] and Lemma 2.8. Moreover, G is 3-edge-connected, although it is not 3-vertex-connected, since removing v 1 and v 2 disconnects the graph. (The graph G does happen to satisfy the zero-three condition, via the divisor (v 1 ) + 2(v 2 ).) Finally, let us determine the automorphism group of G. Any automorphism must send v 2 to v 2 since it is the only vertex of degree 5, and v 1 to v 1 since it is the only vertex not on a cycle of length 3. From there, using adjacency relations of vertices we can determine that Aut(G) is isomorphic to Z/2Z × Z/2Z × Z/2Z, and can be generated by three automorphisms of order 2: one switching u 2 and u 3 , one switching w 2 and w 3 , and one switching u i with w i for all i in {1, 2, 3}. Since |Aut(G)| = 8, the graph G does not have an automorphism of order 3, even though it is 3-edge-connected and has gonality 3.
To see that the zero-three condition is necessary, consider the graph in Figure 4.2, which is the wheel graph W 5 on 5 vertices. It has gonality 3: the divisor (w 1 ) + (h) + (w 3 ) has positive rank; and since the graph has K 4 as a minor, the treewidth of the graph, and thus its gonality, is at least 3 by [6] and Lemma 2.8. It is also 3-vertex-connected. However, we claim that it does not satisfy the zero-three condition. Let D be any effective rank 1 divisor of degree 3 on W 5 . We will see in Lemma 4.2 that D must have support size 1 or 3, and that it is equivalent to some divisor with support size 3. If W 5 satisfies the zero-three condition, then since any three vertices have at least one edge in common, there must be a rank 1 divisor (a) + (b) + (c) on W 5 where a, b, c are distinct and form a K 3 in the graph. It follows that (w i ) + (w i+1 ) + (h) has rank 1 for some i, where addition is done modulo 4. However, this divisor does not have rank 1: starting Dhar's burning algorithm from w i+2 burns the whole graph. Thus W 5 must not satisfy the zero-three condition. Finally, the automorphism group of W 5 is the same as the automorphism group of the cycle C 4 , namely the dihedral group of order 8. This group does not have any elements of order 3. Now, let D be a divisor of degree 3 and rank 1 on a graph of gonality 3. Recall the equivalence relation ∼ D on V (G) resulting from Lemma 3.3: v 1 ∼ D v 2 if and only if v 1 ∈ supp(D v2 ) and v 2 ∈ supp(D v1 ). We will use this relation to define a permutation σ of the vertices of G, which we will then prove to be a cyclic automorphism of order 3. First, we need the following lemma.
Proof. Suppose for the sake of contradiction that there exists a divisor D ∈ Div + (G) such that D ∼ D and Let v 0 ∈ V (G) be distinct from v 1 and v 2 , and start Dhar's burning algorithm at v 0 . Since G is 3-vertex-connected, the graph G − {v 1 , v 2 } is connected, so every vertex in G besides v 1 and v 2 will burn. Since deg(v 2 ) ≥ 3 and since G is simple, there are at least two edges connecting v 2 to G − {v 1 , v 2 }. Both these edges are burning, so v 2 burns since it only has one chip. At this point the whole graph besides v 1 is burning. Since deg(v 1 ) ≥ 3, there are at least three burning edges incident to v 1 .
Since v 1 has two chips, v 1 burns, and thus the entire graph burns. This shows that D is v 0 -reduced. Since v 0 / ∈ supp(D ), D is not a winning divisor, which is a contradiction to D ∼ D.
For each edge from a vertex in [v 1 ] D to a vertex in another equivalence class [u 1 ] D , define σ as follows. If e = v 1 u 1 ∈ E(G) with |supp(D u1 )| = 3, then let σ(u 1 ) = u 2 where u 2 ∈ [u 1 ] D is the unique vertex such that σ(v 1 )u 2 ∈ E(G). Then we must have σ(u 2 ) = u 3 where u 3 is the unique vertex with σ(v 2 )u 3 ∈ E(G). On the other hand, if |supp(D u1 )| = 1, then let σ act as the identity on u 1 .
Let this process, where vertex classes induce orderings on their adjacent vertex classes, propagate outwards. If we reach a situation where a vertex class with one vertex induces an order on a vertex class with three vertices, pick some arbitrary ordering on those three vertices and define σ accordingly. We will show that the order chosen does not matter, and that this process provides us with our desired automorphism.
We are assuming our graph satisfies the zero-three condition, so the map σ mapping v 1 to v 2 to v 3 to v 1 preserves the connectivity of v 1 , v 2 , and v 3 , since either all share edges or none share edges.
Since our graph G is connected, the propagation process induces an order on each vertex class in G. We now argue that we never run into the problem that the induced orderings are incompatible with each other. Suppose for the sake of contradiction that we have a vertex class Let v a , v b , v c be the three vertices in [v] D . First, assume that at least one vertex in [v] D burns, say v a . Since G is 3-vertex-connected, it is still connected after removing v b and v c , so every other vertex in G must burn. We also know that deg(v b ) and deg(v c ) are both at least 3, and so these vertices are adjacent to at least two burning vertices. Since each has one chip, both of these vertices (and thus the entire graph) will burn. This is a contradiction, since r(D) ≥ 1. Having reached a contradiction in all cases, we know that the propagation process can never lead to incompatible orderings. Notice also that if e = uv ∈ E(G), then σ −1 (u)σ −1 (v) ∈ E(G) because σ −1 (u) = σ 2 (u) and σ −1 (v) = σ 2 (v). Hence, we have shown that σ is an automorphism. By definition, σ is cyclic and we have already demonstrated that σ has order 3. Finally, we see that σ does not fix any edge of G because we have already shown that we cannot have an edge between two equivalence classes with one vertex each (recall that the third edge case in Figure 4.3 is impossible).
We may now define the same quotient morphism ϕ : G → G/ ∼ D as in Section 3. However, notice that our equivalence classes of V (G) can now be viewed as orbits under the action of σ. Thus, G/ ∼ D = G/σ. The proof of this proposition will be similar to an argument from the proof of Theorem 3.1, when we showed that the quotient map from G to G/ ∼ D was harmonic of degree 3, and that G/ ∼ D was a tree.
Proof. Since σ is an automorphism, for each vertex v ∈ V (G) such that v ∈ For any given edge [e] D ∈ E(G/σ), Thus, ϕ is a degree 3 morphism. The same argument from the proof of Theorem 3.1 shows that G/σ is a tree.
We are now ready to prove Theorem 4.1.
For (3) =⇒ (2) (for which we will not assume G satisfies the zero-three condition), suppose that there exists a cyclic automorphism σ : G → G of order 3 that does not fix any edge of G, such that G/σ is a tree. We wish to show that ϕ : G → G/σ is a non-degenerate harmonic morphism of degree 3. The argument for this is nearly identical to the argument from Proposition 4.4. However, the proof of harmonicity requires a few additional details. : v ∈ e, ϕ(e) = [e]}| = 3, no matter which edge e we pick. This is because σ fixes v but does not fix any of its incident edges. Hence, these edges must cycle around v with order 3. We have now shown that ϕ is harmonic. By the same computation as in Proposition 4.4, ϕ also has degree 3.
Unlike Theorem 3.1, which only relates divisorial and geometric gonalities, Theorem 4.1 allows us to determine divisorial gonalities using information about graph automorphisms. For example, consider the Frucht graph in Figure 4.4, which is the smallest trivalent graph with no nontrivial automorphisms [16]. It can be computationally verified that the divisor depicted in Figure 4.4 is indeed a winning divisor, so the Frucht graph has divisorial gonality at most 4. It is 3-vertex-connected, and since it has no cyclic automorphisms of order 2 or 3, so either its gonality is 4, or its gonality is 3 and it does not satisfy the zero-three condition. We computationally run through all possible support sets for divisors of degree 3 with exactly 1 or 2 edges between the three vertices, and verify none have positive rank. Thus the Frucht graph has gonality 4. We now ask whether it can be strengthened. In particular, since the condition of being 3-vertex-connected is relatively strong, we might wonder whether a weaker condition, such as being trivalent, is sufficient for Theorem 1.2 to hold. The next result shows that this is not the case. Proof. First we note that a trivalent graph is 3-vertex-connected if and only if it is 3-edge-connected [9,Theorem 5.12], so our graph G is not 3-edge-connected. Since G is also bridgeless, it must be exactly 2edge-connected. This means that there exists some way to partition G into two subgraphs, H 1 and H 2 , connected by exactly two edges, as illustrated in Figure 4.5. Suppose for the sake of contradiction that there exists D ∈ Div + (G) with deg(D) = 3 and r(D) = 1. Then there exists some divisor D ∼ D such that D has exactly two chips on H 1 and one chip on H 2 : we must be able to move at least one chip onto both subgraphs, and since there are only two edges connecting the subgraphs, we can move at most two chips in a single subset firing move. Let v 1 , v 2 ∈ H 1 and v 3 ∈ H 2 be the vertices such that supp(D ) = {v 1 , v 2 , v 3 }. We will split into two cases: first where removing v 1 and v 2 disconnects the graph, and second where removing them leaves the graph connected. We split the former case into two subcases, depending on the relationship of v 1 and v 2 to a connected component of the disconnected graph which does not contain v 3 .
Assume that removing v 1 and v 2 disconnects the graph into at least two connected components. A trivalent 2-edge-connected graph is also 2-vertex-connected, so it follows that v 1 = v 2 . Let H 3 be one of the connected components which does not contain v 3 . This implies that there exists at least one edge incident to both v 1 and some vertex in H 3 , and that the same holds for v 2 . Since each vertex is trivalent, by symmetry, we have at most two edges connecting each vertex in {v 1 , v 2 } with vertices in H 3 .
First we deal with the subcase that there exist at least two edges incident to either v 1 or v 2 entering H 3 , and at least one edge incident to the other vertex entering H 3 . Notice that we are at a state where we cannot fire onto H 3 without inducing debt (see the bottom graph in Figure 4.6). Choose a vertex v 0 ∈ H 3 . Since we are unable to fire without inducing debt, at least one of our two vertices has fewer chips than edges incident to H 3 . Hence, if we begin Dhar's burning algorithm at v 0 , everything in H 3 must burn, including at least one of the two vertices with chips. This forces the other vertex with a chip to burn as well. Since we have only one other vertex with exactly one chip, this implies that the whole graph burns. Now we handle the subcase that there exist exactly one edge incident to v 1 and exactly one edge incident to v 2 entering H 3 . Then there exist two vertices v 1 , v 2 ∈ H 3 which are the endpoints of these edges (see the top graph in Figure 4.6). We know that v 1 = v 2 because G is 2-vertex-connected. Fire onto H 3 , moving the two chips from {v 1 , v 2 } onto {v 1 , v 2 }. Suppose that we can continue firing in this manner, i.e. moving chips onto two vertices which are each connected by exactly one edge to the rest of the graph. Since our graph is finite, this process must terminate at some point. If we are able to hit all vertices in H 3 , we have a contradiction because this implies that at least two vertices in H 3 are not trivalent. Before hitting all of the vertices in H 3 , we reach a state as in the previous case, with at least two edges incident to either of the two vertices entering the subgraph of H 3 that we are unable to fire onto. (Notice that we cannot fire from either vertex separately either, because this would imply the existence of a bridge. ) We initially assumed that r(D) = 1, so we have reached a contradiction. Thus, we know that removing the set {v 1 , v 2 } cannot disconnect the graph. Now we assume that removing v 1 and v 2 does not disconnect the graph. Choose a vertex v 0 ∈ H 2 such that v 0 = v 3 (such a vertex exists due to trivalence). If we begin Dhar's burning algorithm at v 0 , we find that the entirety of H 2 must burn, since there exists only one vertex with a single chip in H 2 . The fire then spreads across the two edges incident to H 1 . Since removing v 1 and v 2 does not disconnect the graph, the fire must burn every vertex in H 1 except possibly v 1 and v 2 . However, because our graph is simple, there exists at most one edge between v 1 and v 2 , implying that each must have at least two incident burning edges. Hence, the whole graph burns, implying that r(D) < 1. Again, this is a contradiction. We conclude that the gonality of the graph is at least 4.
It is worth noting that this result does not extend to multigraphs.   Corollary 4.6. If G is a simple, bridgeless trivalent graph that is not 3-vertex-connected, then ggon(G) = 3.
Proof. By Proposition 4.5, G does not have gonality 3. Notice that arguing that (2) =⇒ (1) in the proof of Theorem 3.1 does not require 3-edge-connectivity. Hence, there exists no non-degenerate harmonic morphism of degree 3 from G to a tree.

Constructing Graphs of Gonality 3
In [8], Chan presents the following construction for all trivalent, 2-edge-connected graphs of gonality 2. Choose a tree T where each vertex v ∈ V (T ) satisfies val(v) ≤ 3.
1. Duplicate T , making two copies T 1 and T 2 .
2. For each vertex v 1 ∈ T 1 with val(v 1 ) ≤ 2, connect it to the matching vertex in T 2 with 3 − val(v 1 ) edges.
Every graph constructed in this way is called a ladder. By [8,Theorem 4.9], each graph arising from this construction has gonality 2, and every 2-edge-connected trivalent graph of gonality 2 with genus at least 3 comes from such a construction.
We now provide a similar construction for graphs of gonality 3. In constrast to the results of [8], not every graph of gonality 3 arises from this construction. For instance, the complete graph on 4 vertices K 4 does not arise from this construction, even though it is 3-vertex-connected and simple with gonality 3; to see this, note that the number of vertices from our construction is always a multiple of 3.
Proposition 5.1. Let T be an arbitrary tree that is not a single vertex, and let S ⊂ V (T ) consist of at least two vertices. Construct a graph T (T ) as follows: 1. Duplicate T twice, for a total of three copies of T . Call these copies T 1 , T 2 , and T 3 .
2. For each vertex v 1 ∈ T 1 with v 1 ∈ S and its corresponding vertices v 2 ∈ T 2 and v 3 ∈ T 3 , connect each pair of vertices with an edge so that all three vertices are connected in a 3-cycle.
In the following proof, we refer to the Cartesian product G H of two graphs G and H. This is the graph with vertex set V (G) × V (H), and an edge between (u, u ) and (v, v ) if and only if u = v and uv ∈ E(G), or u = v and u v ∈ E(H).
Proof. It is clear that the morphism ϕ : T (T ) → T which maps corresponding triples of vertices {v 1 , v 2 , v 3 } to each other is a non-degenerate harmonic morphism. Notice that arguing that (2) =⇒ (1) in the proof of Theorem 3.1 does not require 3-edge-connectivity. Hence, there exists a divisor D on T (T ) such that deg(D) = 3 and r(D) ≥ 1, meaning gon(T (T )) ≤ 3.
Since S contains at least two vertices, the graph T (T ) has K 2 K 3 as a subgraph. This graph in turn has K 4 as a minor, which is the forbidden minor of graphs of treewidth 2 [6]. Thus, gon(T (T )) ≥ tw(T (T )) ≥ 3 by Lemma 2.8. We conclude that gon(T (T )) = 3.  Proof. Choosing S = V (T ) and performing our construction yields T (T ) = T K 3 , which thus has gonality 3.
We can extend our construction to include certain multigraphs. Notice that we can add arbitrary edges between corresponding triples of vertices (which are already connected via a 3-cycle) while retaining a graph of gonality 3. This is because we still have the same non-degenerate harmonic morphism (the added edges are simply contracted) and because treewidth of a multigraph is equal to the treewidth of the underlying simple graph.
We can also generalize this construction somewhat to create graphs of gonality k > 3, although we are more constrained in what set of vertices we can choose for S. Make k copies of a tree T that has at least two vertices. For each vertex v of T with val(v) ≤ k − 1, connect all the k copies of v to each other with k 2 edges. (Including some vertices with val(v) ≥ k is also allowable.) Call the resulting graph T (T ). Our construction guarantees that each vertex has valence at least k, so gon(T (T )) ≥ k by Lemma 2.9. There is a natural harmonic morphism of degree k from T (T ) to T , which by the argument from (2) =⇒ (1) in the proof of Theorem 3.1 shows that gon(T (T )) ≤ k. We conclude that gon(T (T )) = k.

Future Directions
There are many cases of graphs of gonality 3 that have not yet been covered by our results. One could consider multigraphs that are not 3-edge-connected, and simple graphs that are neither 3-vertex-connected nor trivalent. Moreover, all the results results in this paper only hold for combinatorial graphs, as opposed to metric graphs, which have lengths associated to their edges. A natural generalization of our work would be to determine the extent to which our results hold for metric graphs. The work by [8] on hyperelliptic graphs was done in the setting of metric graphs, so some of our results may extend via similar arguments.
Another natural question would be that of algorithmically testing whether or not a graph has gonality 3. In general, computing the divisorial gonality of a graph is NP-hard [18], but it is possible to check if a graph has gonality 2 in O(n log n + m) time [5]. The next step would be to develop an efficient algorithm for determining if a graph has gonality 3. There is a naïve polynomial time algorithm that enumerates all effective divisors of degree 3, then tests each such divisor against all possible placements of −1 chips using Dhar's burning algorithm. However, a more efficient algorithm could be a helpful computational tool. The criteria we present in Theorem 1.2 may be useful for this endeavor.