A Gray code for arborescences of tournaments

A Gra y co de for arb orescences of tournamen ts Marthe Bonam y ∗ Mic hael Hoffmann † Cl ´ emen t Legrand-Duchesne ‡ G ¨ un ter Rote § Marc h 31, 2026 Abstract W e consider the follo wing question of Kn uth: giv en a directed graph G and a root r , can the arborescences of G rooted in r b e listed suc h that an y t wo consecutiv e arb orescences differ by only one arc? Suc h an ordering is called a pivot Gray co de and can b e formulated as a Hamiltonian path in the reconfiguration graph of the arb orescences of G under arc flips, also called flip graph of G . W e giv e a p ositiv e answ er for tournamen ts and explore several conditions sho wing that the flip graph of a directed graph ma y contain no Hamiltonian cycles. A Gr ay c o de is a linear or cyclic order on the elements of a fixed set (usually bit representations of n umbers b et w een 0 and 2 n − 1), such that consecutiv e elemen ts differ on exactly one bit. Gra y co des w ere originally considered to a void the errors in tro duced b y unperfectly sync hronised ph ysical switches, causing a p eriod of transition b et ween t wo consecutive binary num b ers. A Gray co de can also b e seen as a Hamiltonian path or even a Hamiltonian cycle in the reconfiguration graph: the graph whose vertices are the enumerated ob jects, with an edge b etw een an y t w o ob jects at Hamming distance one (see M¨ utze’s extensiv e surv ey on Gra y codes for com binatorial structures [ M ¨ ut23 ]). In the con text of spanning trees of a graph G , t w o t yp es of Gray co des can b e considered. The most general one is an order with the r evolving do or property: eac h spanning tree in the sequence is obtained from the previous one by an edge exc hange, that is, b y remo ving an edge and adding another one. The first such algorithm w as given by Cummins in 1966 [ Cum66 ]. In fact, Cummins sho wed that the reconfiguration graph of spanning trees under edge exc hanges of any graph G , also called flip gr aph of G , is edge-Hamiltonian. Namely , for eac h edge of the flip graph there exists a Hamiltonian cycle passing through this edge. Shank [ Sha68 ] then ga v e a short and simple proof of this result. As of today , the most efficien t kno wn Gray co de en umeration algorithm for spanning trees runs in constant delay b etw een consecutiv e outputs on a verage and w as giv en b y Smith [ Smi97 ]. Finally , the set of spanning trees of a graph G are the bases of the matroid formed by the edges of G and Holzmann and Harary [ HH72 ] generalised ∗ CNRS, LaBRI, Universit ´ e de Bordeaux, F rance ( marthe.bonamy@u-b ordeaux.fr ) † Departmen t of Computer Science, ETH Z ¨ uric h, Switzerland ( hoffmann@inf.ethz.ch ) ‡ Theoretical Computer Science Department, F acult y of Mathematics and Computer Sci- ence, Jagiellonian Universit y , Krak´ ow, Poland ( clement.legrand-duchesne@uj.edu.pl ). § Institut f ¨ ur Informatik, F reie Universit¨ at Berlin ( rote@inf.fu- berlin.de ). 1 Shank’s pro of to the reconfiguration graph of the bases of an y matroid under elemen t exc hanges. The second type of Gra y co des for spanning trees are those with the str ong r evolving do or property , that is, those in which the edges added and deleted at eac h step share a common endpoint. These Gra y codes are also referred to as pivot Gr ay c o de . Problem 1. Do es every gr aph G admit a pivot Gr ay c o de on its sp anning tr e es? Although it is rather simple to prov e that the corresp onding reconfiguration graphs is connected for any graph G , it remains op en whether all graphs ad- mit a pivot Gray co de for spanning trees. Indeed, one of the main tec hniques used to construct a Gra y co de for spanning trees consists in partitioning the spanning trees into t wo sets: those containing a sp ecific edge e and those av oid- ing it. One can then apply induction on b oth sets by considering the graph G/e where e is contracted, and the graph G − e where e is remov ed, resp ec- tiv ely . How ever, the uncontraction of the edge e do es not preserv e the strong rev olving do or property , as the edges inciden t to e are adjacent in G/e but not in G . Nev ertheless, pivot Gra y codes for spanning trees ha ve been constructed in some graph classes, namely for fan graphs [ CGS24 ] and more generally for outerplanar graphs [ BM24 ]. As Kn uth noted [ Kn u11 , Answ er to Exercise 7.2.1.6–102], Problem 1 would immediately follo w from the existence of a Gra y co de on the ro oted arb ores- cences of a directed graph, b y replacing each edge by t wo arcs going in b oth directions. More precisely , an arb or esc enc e of a directed graph G ro oted in r is a spanning tree of G directed a w ay from r . Given t wo arb orescences that differ in exactly one arc, the arcs added and deleted must p oin t to the same v ertex u , for this vertex u to b e accessible in b oth arb orescences from r . Therefore, Gra y co des on the arb orescences of G naturally hav e the strong rev olving door prop ert y . The problem of designing such a Gray co de w as proposed by Knuth, with an estimated difficult y of 46/50: Problem 2 (Exercise 7.2.1.6–102 in [ Knu11 ]) . Do es every dir e cte d gr aph G admit a pivot Gr ay c o de on its arb or esc enc es r o ote d in a given vertex? In 1967, Chen [ Che67 , Theorem 1] claimed that for an y directed graph G and v ertex r ∈ V ( G ), the flip graph on the arb orescences of G ro oted in r con- tains a Hamiltonian cycle, provided there are at least three arb orescences. A coun terexample to this claim was pro vided by Rao and Ra ju [ RR72 ] in 1972: they constructed a family of directed graphs whose flip graph is a path. These digraphs are not just a sp oradic exception: w e characterise the cases where an arborescence has degree 1 in the flip graph. In Section 2.2 , w e construct a greater v ariet y of counterexamples of directed graphs with unbalanced bipar- tite flip graphs, hence without Hamiltonian cycle. Ho wev er, as we will show, the im balance for these coun terexamples cannot exceed one; thus they do not con tradict the existence of a Hamiltonian p ath in the flip graph. Finally , our main result is the construction of a piv ot Gra y co de on the arb orescences of any tournamen t (see Section 3 ): Theorem 1. L et G b e a tournament and r b e a vertex of G . The flip gr aph of the arb or esc enc es of G r o ote d in r admits a Hamiltonian p ath. 2 1 Preliminaries 1.1 Notations and glossary Giv en a directed graph G = ( V , E ), w e denote b y u → v or uv an arc going from u to v . W e denote b y N + ( v ) = { u : v → u } the outneighb ourho o d of v and deg + ( v ) = | N + ( v ) | its outde gr e e . Analogously , we denote by N − ( v ) = { u : u → v } its inneighb ourho o d and deg − ( v ) = | N − ( v ) | its inde gr e e . A v ertex of outdegree zero is a sink . The supp ort of a directed (multi)-graph G is the undirected graph on the same vertex set, with an edge b etw een u and v if G has at least one arc from u to v or vice v ersa. Giv en a fixed v ertex r of G called r o ot , we call arb or esc enc e any spanning directed tree of G in which all arcs are oriented a wa y from r . Tw o arb orescences that are equal on all arcs but one are said to differ by an ar c flip . Flipping in the arc uv in an arb orescence T corresp onds to remo ving the unique arc entering v and replacing it by the arc uv , thereby obtaining another arb orescence that differs from T by an arc flip. The flip gr aph of G ro oted in r , is the undirected graph F r ( G ) whose no des are the arb orescences of G ro oted in r , with an edge betw een eac h pair of arborescences that differ by an arc flip. Giv en a directed graph G ro oted in r and one of its vertices u , denote D G ( u ) the desc endants of u , that is the v ertices v such that all paths from r to v pass through u ( u included). Given an arb orescence A of G , an arc uv ∈ E ( G ) \ E ( A ) can b e flipp ed in if and only if v is not a descendant of u in A . W e no w prov e an auxiliary lemma allo wing us to extend subarb orescences in to arborescences. Lemma 2. L et G b e a dir e cte d gr aph and r a vertex of G , such that G admits at le ast one arb or esc enc e r o ote d in r . Any dir e cte d subtr e e T of G r o ote d in r c an b e c omplete d into an arb or esc enc e. Pr o of. W e pro ceed b y induction of | V ( T ) | . Let A b e an arb orescence of G and T a directed subtree of G , b oth rooted in r . Let X b e the set of v e rtices of G that do not b elong to T . Let P b e a minimal path from r to a vertex in X , suc h a path exists b ecause A is an arb orescence ro oted in r . Let v and u b e the last and p en ultimate v ertices in P . W e hav e u → v and u b elongs to V ( T ) by minimalit y of P ; therefore, adding the arc uv to T extends T . 1.2 Hamiltonicit y of ladders and h yp ercub es W e start with some easy constructions of Hamiltonian paths in ladders and h yp ercub es, whic h will be used in other constructions. A ladder of length n is the undirected graph consisting of t w o paths a 1 , . . . , a n and b 1 , . . . , b n of length n , with an additional edge betw een a i and b i for all i . A v ertex in the ladder has level i if v ∈ { a i , b i } . Lemma 3. L et F b e the ladder of length n , and let i, j ≤ n b e inte gers with i  = j . L et u b e a vertex of level i . Then F has a Hamiltonian p ath fr om u to some vertex at level j . Note that w e cannot prescrib e which of the t wo vertices at level j is the endp oin t of this Hamiltonian path. In fact, since F is bipartite, a parity argu- men t sho ws that only one of the t w o v ertices at lev el j can potentially b e the endp oin t of a Hamiltonian path starting at u . 3 Pr o of. Without loss of generality , assume that j > i and u = a i . Consider the paths P 1 , P 2 and P 3 , where P 1 = a i , . . . , a 1 , b 1 , . . . , b i and P 3 = b j , . . . , b n , a n , . . . , a j , and P 2 go es from b i to some vertex v ∈ { a j , b j } b y zigzagging (see Figure 1 ): P 2 = b i , b i +1 , a i +1 , a i +2 , b i +2 , b i +3 , . . . , v Concatenating the three paths P 1 , P 2 , and P 3 results in a Hamiltonian cycle, ending at a vertex w  = v of level j . Note that w dep ends on the parity of j − i . u v w u w v Figure 1: Tw o Hamiltonian paths in a ladder with extremities at differen t lev els. The paths P 1 and P 3 are dra wn in blue, the path P 2 in red. The hyp er cub e K d 2 of dimension d is the graph on 2 d v ertices indexed by { 0 , 1 } d in whic h t wo v ertices are adjacen t if they differ on exactly one co ordinate. Hyp ercub es of dimension at least 2 are edge-Hamiltonian: Lemma 4. L et K d 2 b e the hyp er cub e of dimension d ≥ 2 and uv ∈ E ( G ) . The hyp er cub e K d 2 has a Hamiltonian cycle c ontaining the e dge uv . Listing binary strings w as the original problem considered b y Gray [ Gra53 ], and thus, many differen t proofs of the Hamiltonicit y of the h yp ercub e exist, such as the binary reflected Gray co de. F rom this, one can deduce that the hypercub e is edge-Hamiltonian simply b y the edge-transitivit y of the h yp ercub e. 1.3 Reductions W e now prov e several reductions rules that preserve the flip graph of a directed graph, or at least the existence of a Hamiltonian path in it. The operations w e consider are the remo v al, the subdivision, the con traction and the duplication of an arc. Arc deletion. Giv en a fixed directed graph G , we denote G − uv the directed graph obtained by removing the arc u → v . Observ ation 5. L et uv b e an ar c of a dir e cte d gr aph G r o ote d in r . The flip gr aph of G − uv is isomorphic to the sub gr aph of F r ( G ) induc e d by the arb or esc enc es not c ontaining uv . A dir e ct c onse quenc e of this is that every ar c that app e ars in none of the arb or esc enc es of G c an b e r emove d fr om G without affe cting its flip gr aph. 4 Let G b e a directed graph G ro oted in some vertex r . W e will sa y that a directed graph H is built on G if V ( H ) = V ( G ) and E ( G ) ⊆ E ( H ), and for ev ery arc u → v in E ( H ) \ E ( G ), u is a descendant of v in G . W e will call suc h an arc u → v a b acke dge of H . Lemma 6. If H is built on G , then F r ( H ) is isomorphic to F r ( G ) . Pr o of. The arcs uv of E ( H ) \ E ( G ) do not app ear in any arborescence of H . Otherwise, let A b e an arb orescence of H with uv ∈ E ( A ). Without loss of generalit y , assume that uv was c hosen such that uv is the only arc of E ( A ) \ E ( G ) on the path from the ro ot r to v . Then there exists a path from r to u that uses arcs of G and av oids v , which contradicts the fact that u is a descendan t of v in G . Thus the arborescences of H are exactly the arb orescences of G , and the flip graphs of H and G are isomorphic b y Theorem 5 . Arc contraction. Denote G/uv the directed graph obtained b y con tracting the arc u → v , that is replacing u and v b y a v ertex w with N − ( w ) = N − ( u ) ∪ N − ( v ) \ { u, v } and N + ( w ) = N + ( u ) ∪ N + ( v ) \ { u, v } . Arc con tractions do not b eha v e as w ell as edge deletions with resp ect to the flip graph. Indeed, the flip graph can change when con tracting an arc, even if this arc b elongs to all the arb orescences of G . F or example, the graph G on three v ertices r , u and v , ro oted in r and containing the arcs r u , uv and r v , has t wo arborescences: one con taining r u and uv , the other con taining r u and r v . Ho wev er con tracting r u results in a single arc, hence the num b er of arb orescences of G was not preserv ed. Recall also from the in tro duction that another problem might o ccur. The reconfiguration sequences on the contracted graph do not corresp ond to reconfiguration sequences in the original graph. F or example, consider the graph G on four vertices r , x , y , z , rooted in r , with four arcs r → x , r → y , x → z and y → z (see Figure 2a ). When con tracting xz into a single vertex u , the arc r u can b e flipp ed to y u (see Figure 2b ). Ho wev er, r and y hav e disjoin t outneigh b ourho o ds in G so this flip do es not corresp ond to a flip in the original graph. r x y z (a) G r y u (b) G/xz Figure 2: A directed graph G for whic h con tracting xz into u creates a flip unfeasible in G . Throughout this article, we will consistently represen t the ro ot of our directed graphs b y a circled v ertex, here r . Ho wev er, some con tractions still preserve the existence of a Hamiltonian path. Let G be a directed graph, r its root and r x an outgoing arc of r . Let G ′ = G/r x b e the directed graph obtained after the con traction of rx into the new ro ot r ′ . Let T /rx denote the set of arb orescences of G that contain the arc r x . 5 Lemma 7. L et A ′ b e an arb or esc enc e of G ′ . If F r ′ ( G ′ ) c ontains a Hamiltonian p ath starting fr om A ′ , then F r ( G )[ T /rx ] c ontains a Hamiltonian p ath starting fr om any arb or esc enc e A such that c ontr acting r x r esults in A ′ . Pr o of. Let ϕ b e the map that asso ciates to an y arc uv of G the arc r ′ v if u ∈ { r, x } and the arc uv otherwise. F or ev ery arb orescence T ro oted in r and con taining the arc r x , w e hav e E ( T /r x ) = ϕ ( E ( T )). Let T ′ b e an arb orescence of G ′ . Let M T ′ b e the set of vertices in N + G ( x ) ∩ N + G ( r ) that are outneighbours of r ′ in T ′ , and let H T ′ b e the set of all arb ores- cences T of G ro oted in r that con tain the arc r x and suc h that ϕ ( T ) = T /r x = T ′ . W e first prov e that H T ′ induces a hypercub e of dimension | M T ′ | in F r ( G ). F or ev ery arc r ′ z ∈ E ( T ′ ) with z ∈ M T ′ , the arcs mapp ed b y ϕ to r ′ z are r z and xz . F or an y other arc in T ′ , there is a unique arc in G mapp ed to it b y ϕ . Hence, H T ′ is in one-to-one corresp ondence with the subsets of M T ′ . Moreov er, t wo arb orescences T 0 and T 1 in H T ′ differ by a flip if only if there exists some z ∈ M T ′ suc h that r z ∈ T i and xz ∈ T 1 − i , and T 0 − z = T 1 − z . Therefore, H T ′ induces in F r ( G ) a subgraph isomorphic to the Hasse diagram of the p oset ( M T ′ , ⊂ ), i.e. a h yp ercub e of dimension | M T ′ | . W e no w show that for every S ′ , T ′ adjacen t in F r ′ ( G ′ ), for every arb orescence S in H S ′ , there is a path in F r ( G )[ T /rx ] starting from S that visits exactly H S ′ b efore ending at some arb orescence in H T ′ . The lemma then follo ws from applying this iterativ ely on the edges of the Hamiltonian path of F r ′ ( G ′ ). Let S ∈ H S ′ . As H S ′ induces a hypercub e of dimension | M S ′ | in F r ( G ), it con tains a Hamiltonian path starting at S and ending at some S 2 ∈ H S ′ . Let a ′ b ′ b e the arc flipp ed in in S ′ to obtain T ′ . Note that b ′ cannot b e equal to r ′ , b ecause r ′ is the ro ot of G ′ , so b ′ is also a vertex of G . If a ′ = r ′ , then by definition of con traction, there exists a ∈ { r, x } such that b ′ ∈ N + G ( a ). Otherwise, a ′ w as not the result of the con traction and we set a := a ′ . In b oth cases, the path from r ′ to a ′ in S ′ a voids b ′ , b ecause the arc a ′ b ′ could b e flipped in to obtain T ′ , so there is also a path from r to a a voiding b ′ in S 2 b ecause b ′ / ∈ { r, x } . Th us, a is not a descendant of b ′ in S 2 and the arc ab ′ can b e flipp ed in in S 2 . Denote T the resulting arb orescence. As ϕ ( ab ′ ) = a ′ b ′ , we ha ve T /r x = T ′ , whic h concludes the pro of. Arc sub division. The follo wing observ ation sho ws that w e can restrict Prob- lem 2 to orien ted graphs b y sub diving one arc in each bigon of some directed graph. Observ ation 8. Sub dividing an ar c do es not mo dify the flip gr aph. Pr o of. Let G b e a directed graph rooted in r , let G ′ b e the directed graph ob- tained by sub dividing an arc uw , and denote v the v ertex introduced in the op eration. W e build a bijection ϕ from the arb orescences of G to the arb ores- cences of G ′ . Giv en an arborescence A containing uw , let ϕ ( A ) b e the arb ores- cence of G ′ obtained b y sub dividing uw by introducing the v ertex v . Given an arb orescence A that do es not con tain uw , let E ( ϕ ( A )) = E ( A ) ∪ { uv } . It is clear that ϕ is a bijection and that ϕ preserv es the Hamming distance, i.e. dist F r ( G ′ ) ( ϕ ( A 1 ) , ϕ ( A 2 )) ≤ dist F r ( G ) ( A 1 , A 2 ) for every arb orescences A 1 and A 2 . Moreo ver, note that all arb orescences of G ′ con tain the arc uv b ecause v has indegree one, thus ϕ − 1 also preserv es the Hamming distance, which prov es that F r ( G ) and F r ( G ′ ) are isomorphic. 6 Arc duplication. The following lemma shows that generalising Problem 2 to directed m ulti-graphs does not increase its difficult y . Lemma 9. L et G b e a dir e cte d multi-gr aph r o ote d in r . L et e b e an ar c of G and let G ′ b e the dir e cte d multi-gr aph obtaine d fr om G by duplic ating e , i.e. adding an ar c e ′ with same he ad and tail as e . If F r ( G ) admits a Hamiltonian p ath (or cycle), then so do es F r ( G ′ ) . Pr o of. F or every arb orescence A that contains e , denote by A ′ the arb orescence obtained from A b y flipping in e ′ . Let A and B be tw o arb orescences adjacen t in F r ( G ) by flipping in an arc f . If b oth A and B con tain e , then { A, A ′ , B , B ′ } induces in F r ( G ′ ) a four-cycle: A and A ′ (resp. B and B ′ ) are adjacent by flipping e in to e ′ and A ′ is adjacen t to B ′ b y flipping in the arc f . If only A con tains e , then f = e , so { A, A ′ , B } induces a triangle in F r ( G ′ ) b ecause if e can b e flipped in in B , then e ′ can also be flipped in. By replacing the edges of the Hamiltonian path (or cycle) P = ( A 1 , . . . A p ) of F r ( G ) b y appropriate paths through triangles or 4-cycles dep ending on whether A i and A i +1 con tain e , one can build a Hamiltonian path P ′ (or cycle) of F r ( G ′ ) such that, whenever P visits an arb orescence A i with e ∈ A i , P ′ visits A i and A ′ i (see Figure 3 ). Figure 3: Constructing a Hamiltonian path of F r ( G ′ ) from a Hamiltonian path of F r ( G ). In the Hamiltonian path of F r ( G ) on the left, the vertices that corresp ond to arb orescences con taining e are dra wn in red. On the righ t a Hamiltoninan path of F r ( G ′ ). The vertices corresponding to arb orescences con- taining e are dra wn in red, those con taining e ′ in blue. The dotted edges are edges that are presen t in the flip graph, but not used in the Hamiltonian path. 2 Directed graphs with no Hamiltonian cycle in their flip graph In this section, we giv e sev eral coun terexamples witnessing that the flip graph of a directed graph ma y not contain a Hamiltonian cycle. 7 2.1 Flip graph can b e paths In [ RR72 ], Rao and Ra ju show ed that the flip graph of every bidirected cycle is a path. W e recall their pro of here: Lemma 10 ([ RR72 ]) . L et G b e the dir e cte d gr aph obtaine d by r eplacing e ach e dge of a n -cycle by a bigon. The flip gr aph of G is a p ath on n vertic es. Pr o of. Let V = { v 1 , . . . , v n } b e the vertices of G , such that v 1 is the ro ot and each v ertex v i is incident to v i − 1 and v i +1 , where the indices are considered mo dulo n . There are only tw o paths form v 1 to v i in G : the first one is v 1 , . . . , v i and the second v 1 , v n , . . . , v i . So every arb orescence of G ro oted in r corresp onds to a unique in terv al [1 , i ] of indices such that v j is reached through a path using the arc v 1 v 2 for eac h j ∈ [1 , i ]. In eac h of these arb orescences, only the arcs en tering the leafs can b e swapped. Thus the reconfiguration graph of G is path on n vertices (see Figure 4 ). Figure 4: The flip graph of a bidirected 5-cycle 2.2 Un balanced bipartite flip graphs Theorem 11. L et G b e a dir e cte d gr aph r o ote d in r , in which al l vertic es have inde gr e e at most two. Then the flip gr aph F r ( G ) is bip artite. Pr o of. W e can express the parit y of the arborescences as a m ultiplicative weight , as follo ws: If xz and y z are tw o incoming arcs of the same v ertex z in G , we arbitrarily assign w eigh t w wz = +1 to one of these arcs and w y z = − 1 to the other arc. Every arc that is the single incoming arc of a v ertex gets weigh t +1. The w eigh t w ( A ) ∈ { +1 , − 1 } of an arborescence A is then the product of the w eights of its arcs. Flipping an arc changes the sign of this weigh t, and thus the w eigh t defines a bipartition of the flip graph. If the tw o bipartition classes differ in size by at least one, then this constitutes y et another coun terexample of a directed graph without a Hamiltonian cycle in its flip graph. F or example, the graph in Figure 5 has a bipartite flip graph with thirteen vertices, which is shown in Figure 6 . W e will also see another example with sev en arborescences in the pro of of Theorem 1 , see Figure 8 . If the tw o bipartition classes differ in size by at least 2, this w ould imme- diately b e an obstacle to the existence of a Hamiltonian path in the flip path. Ho wev er, w e can pro v e that in the flip graph of a directed graph with indegree at most t wo, the size of the tw o parts of the bipartition differs by at most 1 (and it is easy to test whether they differ b y 1, i.e. the total num b er of trees is o dd). Theorem 12. L et G b e a dir e cte d gr aph r o ote d in r , in which al l vertic es have inde gr e e at most two. The two bip artition classes differ in size by at most 1. 8 1 2 3 4 5 L =       0 1 1 1 0 0 0 − 1 0 0 0 0 0 − 1 1 0 0 0 0 − 1 0 − 1 0 0 0       Figure 5: A graph with 13 arb orescences and no Hamiltonian cycle in its flip graph. On the righ t, its w eighted Laplace matrix (defined in Theorem 12 ), where the i -th ro w and column corresp ond to the vertex lab elled i . Figure 6: The flip graph of the graph drawn on Figure 5 Pr o of. Denote w ( A ) the weigh t function defined in the pro of of Theorem 11 . W e claim that      X A w ( A )      ≤ 1 . The sum P A w ( A ) ov er all arb orescences A expresses the difference b etw een the n um b er of “p ositive” and “negativ e” arb orescences. This sum can b e calculated by the w eighted matrix-tree theorem (see for example [ CK78 ]), which says the following: Let L = ( L ij ) b e the w eigh ted Laplace matrix, which has entry L ij = − w ij for eac h arc i → j , and L ij = 0 9 for all off-diagonal en tries that don’t corresp ond to an arc. The diagonal entries L ii are c hosen to mak e the column sums zero. Let ˜ L b e the matrix L after remo ving the ro w and column corresp onding to the ro ot. Then X A w ( A ) = det ˜ L. In our case, the matrix L lo oks as follo ws: F or a vertex with t w o incoming arcs, the corresp onding column of L has t w o entries +1 and − 1, and hence the diagonal en try in this column, which is supp osed to balance the column sum, is zero. F or a vertex with a single incoming arc, the corresponding column of L has an entry +1, and hence the diagonal entry is − 1. The matrix ˜ L has therefore the follo wing properties. • All en tries are 0, +1, or − 1. • There are at most t w o nonzeros per column. • If a column has t w o nonzeros, they are of opposite sign. The matrices arising from net work flo w problems hav e the same prop erties. The Theorem of Heller and T ompkins [ HT56 ] c haracterises the totally unimo dular matrices among the matrices with the first tw o prop erties, and it implies in this case that ˜ L is totally unimo dular, and in particular, it has determinant 0 or ± 1. It is easy to see this directly: If there is a zero column, the determinan t is zero. If there is a column with a single nonzero, w e expand the determinan t b y this column, and w e obtain a smaller matrix that fulfills the same conditions. Rep eating this process, we either find a zero column at some point, or we reduce the matrix to a trivial matrix of size 1 × 1, or to a matrix where eac h column con tains a +1 and a − 1. In the first and third cases, det ˜ L = 0; in the second case, det ˜ L = ± 1. The pro cedure can easily be carried out com binatorially . Deleting the ro w of the ro ot means that all neighbours of the root lose an incoming arc. A vertex j with a single incoming arc corresponds to a column j with a single nonzero en try , sa y L ij . Expanding this column deletes row i and column j . In the graph, this corresp onds to remo ving all edges out of i , which means that the outneighbours of i will ha ve their indegree reduced, and the pro cess can contin ue. If the process stops b efore disman tling the whole graph, the determinan t is zero, otherwise it is ± 1. 2.3 Arb orescences of degree one in the flip graph Another prop ert y preven ting the existence of Hamiltonian cycles in the flip graph is the existence of degree-one v ertices, whic h migh t o ccur ev en when the flip graph is not bipartite (and th us not a path). F or example, the graph drawn on Figure 7 is an oriented graph with this prop erty . The degree-one vertices in the flip graph can b e c haracterised as follows: Observ ation 13. L et G b e a dir e cte d gr aph r o ote d in r and A an arb or esc enc e of G such that A has de gr e e one in the flip gr aph F r ( G ) , by flipping in some ar c uv . Then G − uv is built on A . 10 Figure 7: An orien ted graph and its flip graph, which contains a degree-one v ertex Pr o of. Let xy ∈ E ( G − uv ) \ E ( A ). Assume to wards contradiction that x / ∈ D A ( y ). So in A , the arc xy can b e flipp ed in as x is accessible from r without passing through y in A . Thus xy = uv , a con tradiction. 3 A Gra y co de for arb orescences in tournamen ts W e pro ve the follo wing refinemen t of Theorem 1 : Theorem 14. L et G b e a dir e cte d multi-gr aph and r b e a vertex of G such that the supp ort of G − r is a clique (in p articular, G c an have ar cs b etwe en two vertic es u and v , p ossibly in opp osite dir e ctions). Then the flip gr aph F r ( G ) is empty or has a Hamiltonian p ath. First, w e observe that Theorem 14 is optimal in the sense that there exist tournamen ts with a flip graph that con tains a Hamiltonian path but no Hamil- tonian cycle. The graph G drawn on Figure 8 with its flip graph is an example of suc h tournamen t, and note also that G falls into the conditions of Theorem 11 . The rest of this section consists in the pro of of Theorem 14 . W e first observ e that by Theorem 9 , it suffices to prov e Theorem 14 for directed graphs without m ultiple arcs. Let G denote the set of pairs ( G, r ) where G is a directed graph, r is a ro ot in G and the supp ort of G − r is a clique. Note that G is stable under the follo wing operations: • deletion of arcs inciden t to r : ∀ u, ( G − r u, r ) ∈ G • con traction of arcs r u inciden t to r : let G ′ b e the graph obtained from G b y remo ving r and u and replacing them b y a vertex w with arcs wx for all x ∈ N + ( r ) ∪ N + ( u ). Then ( G ′ , w ) ∈ G 11 Figure 8: An orien ted graph with seven arb orescences and whose flip graph is bipartite, so without Hamiltonian cycle. W e pro ceed b y induction. Let ( G, r ) ∈ G . The arcs ending in r do not belong to an y arborescence, so by Theorem 5 , w e assume without loss of generalit y that N − ( r ) = ∅ . If r has only one outneighbour u , then the arc r u is contained in ev ery arb orescence of G ro oted in r . Hence, by Theorem 7 , F r ( G ) is isomorphic to F u ( G − r ) and b y induction b oth admit a Hamiltonian path or are empt y . If r has at least t wo outneigh b ours u and v , then they are connected b y an arc, sa y u → v , b ecause the supp ort of G − r is a clique. The vertices u and v migh t also be connected b y the arc v → u , but we do not care at this stage of the proof. W e c ho ose u and v as follows: among all pairs ( u, v ) ∈ N + ( r ) 2 with uv ∈ G , w e c ho ose one suc h that u has an outneighbourho o d of maximal size. The set of arb orescences of G can be partitioned in to four typ es of arb ores- cences: those that do not contain the arc e = r u , those that con tain r u and f = r v , those that contain r u and g = uv and those that r u but neither r v nor r v . These t yp es partition F r ( G ) in to four subsets, that we will denote T − e , T /e/f , T /e/g and T − f − g /e resp ectiv ely . 3.1 Structure of each type Lemma 15. If the flip gr aph of G r o ote d in r is non-empty, the differ ent typ es of arb or esc enc es induc e the fol lowing structur es in F r ( G ) : • F r ( G )[ T − e ] is empty or c ontains a Hamiltonian p ath P − e , • F r ( G )[ T − f − g /e ] is empty or c ontains a Hamiltonian p ath P − f − g /e , • F r ( G )[ T /e/f ∪ T /e/g ] c ontains a sp anning ladder. Pr o of. As a direct application of Theorem 5 , we get that F r ( G )[ T − e ] is iso- morphic to F r ( G − e ), so by induction F r ( G )[ T − e ] is empty or also admits a Hamiltonian path P − e . 12 Using Theorems 5 and 7 , w e get that F r ( G )[ T − f − g /e ] is isomorphic to F w ( H ), where H is obtained from G b y con tracting e in to w and deleting w v . By induction, F r ( G )[ T − f − g /e ] is empty or admits a Hamiltonian path P − f − g /e . Finally , note that F r ( G )[ T /e/f ] and F r ( G )[ T /e/g ] are isomorphic via flipping the arc f in to the arc g . They are also isomorphic to F w ( H ), where H is obtained from G b y contracting e and g into w . Moreov er, note that F r ( G )[ T /e/f ∪T /e/g ] is not empt y , b ecause H admits an arb orescence if and only G do es, by Theorem 2 applied to the subtree of G comp osed of the edges e and g . Combining all these argumen ts with Theorem 7 , we deduce that F r ( G )[ T /e/f ∪ T /e/g ] con tains a spanning ladder (see Figure 9 ). P − e ⊂ F r ( G )[ T − e ] P − f − g /e ⊂ F r ( G )[ T − f − g /e ] F r ( G )[ T /e/f ∪ T /e/g ] Figure 9: Spanning structures within the differen t t yp es of arb orescences. 3.2 Assem bling the pieces Pr o of of The or em 14 . W e first argue that one can assume without loss of gen- eralit y that T − f − g /e and T − e are non-empty . If both are empt y , then by The- orem 15 F r ( G ) = F r ( G )[ T /e/f ∪ T /e/g ] contains a spanning ladder and Theo- rem 14 follo ws directly from Theorem 3 . If T − f − g /e is empt y and T − e is not, then F r ( G )[ T − e ] con tains a Hamiltonian path P − e b y Theorem 15 . By flipping in e in one of the extremities of P − e , one obtains an arborescence in T /e/f ∪ T /e/g . By Theorem 15 F r ( G )[ T /e/f ∪ T /e/g ] con tains a ladder, and hence contains a Hamiltonian cycle, whic h can b e used to extend P − e in to a Hamiltonian path of F r ( G ). Similarly , w e can assume that T − e is non-empt y . F or A in T − f − g /e , let A ′ and A ′′ b e the arb orescences in T /e/f (resp ectiv ely T /e/g ) obtained from A b y flipping in f (resp ectiv ely g ). Assume for no w that the extremities A 1 and A 2 of P − f − g /e ha ve distinct A ′ i . By Theorem 15 , F r ( G )[ T /e/f ∪ T /e/g ] contains a ladder, so b y Theo- rem 3 , it con tains a Hamiltonian path going from A ′ 1 to A ′ 2 or A ′′ 2 . Along with P − f − g /e (obtained from Theorem 15 ), one obtains a Hamiltonian cycle of F r ( G )[ T /e/f ∪ T /e/g ∪ T − f − g /e ]. Finally , all arb orescences in T − e are adjacent to some arb orescence in T /e/f ∪ T /e/g ∪ T − f − g /e b y flipping in the edge e , so F r ( G ) con tains a Hamiltonian path starting in T − e (see Figure 10 ). 13 A 1 A 2 P − e ⊂ F r ( G )[ T − e ] P − f − g /e ⊂ F r ( G )[ T − f − g /e ] F r ( G )[ T /e/f ∪ T /e/g ] Figure 10: Hamiltonian path if A ′ 1  = A ′ 2 . Th us we can assume without loss of generality that the extremities A 1 and A 2 of P − f − g /e ha ve A ′ 1 = A ′ 2 (and A ′′ 1 = A ′′ 2 ). As A ′ i is obtained from A i b y flipping in the arc f , this implies that A 1 and A 2 only differ on one arc en tering v , so the path P − f − g /e from Theorem 15 is in fact a Hamiltonian c ycle of F r ( G )[ T − f − g /e ]. Either A ′ = A ′ 1 for ev ery arborescence A in T − f − g /e or there are tw o arb orescences A and B , consecutiv e on P − f − g /e , such that A ′  = B ′ . In the second case, F r ( G )[ T − f − g /e ] con tains a Hamiltonian path whose extremities are A and B , so we are bac k in the case of the previous paragraph. Th us we can assume that for every A ∈ T − f − g /e , w e ha ve A ′ = A ′ 1 and A ′′ = A ′′ 1 . W e no w c haracterise the structure of tournaments for whic h this is p ossible: Lemma 16. L et G b e a dir e cte d gr aph r o ote d in a vertex w , such that the supp ort of G − w is a clique. L et v ∈ V ( G ) such that flipping in w v in any arb or esc enc e r o ote d in w that do es not c ontain wv , r esults in the same arb or esc enc e A . Then the fol lowing hold: (1) F w ( G − w v ) is a clique, (2) al l vertic es other v have a single p ath fr om w to them in G − w v , (3) al l p aths going fr om w to v in G − w v have length at le ast 2, (4) G − w v has the fol lowing structur e: (a) If F w ( G − w v ) is a single vertex, then G − w v is built on the dir e cte d p ath w , v 1 , . . . , v n , with v = v k for some fixe d k > 1 . In other wor ds, G c ontains the gr aph L k,n (dr awn on Figur e 11a in black) as a sub- gr aph, and the ar cs in E ( G ) \ E ( L k,n ) ar e either ending at w or of the form v i +1 → v i (dr awn on Figur e 11a in r e d). (b) If F w ( G − w v ) is a clique on at le ast two vertic es, then G − w v is built on the dir e cte d p ath w , v 1 , . . . , v n with an additional vertex v and al l ar cs v i v with i ≥ k − 1 for some fixe d 1 < k ≤ n . In other wor ds, G c ontains the gr aph M k,n (dr awn on Figur e 11b in black) as 14 a sub gr aph, and the ar cs in E ( G ) \ E ( M k,n ) ar e either v → v k − 1 , or ending at w , or of the form v i +1 → v i (dr awn on Figur e 11b in r e d). w v 1 v 2 v 3 = v v 4 (a) Illustration of (4a) with n = 4 and k = 3 w v 1 v 2 v 3 v 4 v (b) Illustration of (4b) with n = 4 and k = 3 Figure 11: The structure of tournaments G , suc h that all arb orescences of G − w v ro oted in w pro duce the same tree when flipping in w v . The arc w v is dra wn in blue, the blac k arcs are present in G while the red arcs are optional. Pr o of of The or em 16 . All arb orescences of G − w v ro oted in w hav e in common all arcs but the one en tering v b ecause they result in the same arb orescence when flipping in w v . Thus F w ( G − w v ) is a clique, whic h pro v es (1) . P erform a BFS from w in G − w v . If each la y er of the BFS contains only one v ertex, then G − w v consists of a directed path w , v 1 , . . . , v n and some bac k edges, in other words G − w v is built on a directed path. In fact, G con tains all arcs v i v j with i > j + 1 be cause G − w has complete support (see Figure 11a ). The additional p ossible bac kedges are either ending at w , or arcs of the form v i +1 → v i , whic h pro ves (4a) . Moreo v er, note that G − w v has only one arb orescence ro oted in w . If some lay er of the BFS contains more than one vertex, assume tow ards con tradiction that some v ertex x different from v has tw o inneigh b ours with depth no larger than that of x (i.e. either in the same la yer or the previous one). Th us, there are at least t wo paths from w to x in G − w v . By Theorem 2 , eac h of these paths can b e completed in to an arb orescence of G − w v ro oted in w . These t wo arborescences hav e different arc en tering x , hence flipping in wv results in differen t arb orescences, a contradiction. So all la yers are either reduced to one single vertex, or contain t w o vertices v k and v and v k / ∈ N + G ( v ). Hence all v ertices but v hav e at most one inneigh b our at smaller depth in the BFS. Let k b e the depth of v . All other vertices at depth at least k are inneighbours of v b ecause G has complete supp ort. Hence G − w v con tains as a spanning subgraph the path w, v 1 , . . . , v n with one additional v ertex v such that v i → v if i ≥ k − 1. In addition, since G − w has complete supp ort, G − w also con tains all back edges of the form v j → v i with i > j + 1, or v → v i with i < k − 1. In other w ords, the graph M k,n dra wn in black on Figure 11b is a subgraph of G − w v . Finally , G can also contain other back edges, namely v → v k − 1 , the arcs ending at w , or the arcs of the form v i +1 → v i , whic h pro v es (4b) . 15 Note that in b oth cases, v  = v 1 b ecause w v / ∈ G − w v , so k > 1, whic h implies (3) . Moreo ver, as bac k edges by definition never b elong to a path from the root to a v ertex, all vertices but v hav e only one path from w to them in G − w v , whic h pro v es (2) . By Theorem 16 applied to G/e with w b eing the contraction of r u and v := v , this implies that G/e has the structure describ ed in (4) for some k and n and that F r ( G )[ T − f − g /e ] is a clique (b y (1) ). F or ev ery B ∈ F r ( G )[ T − e ], let ˜ B b e the arb orescence of F r ( G )[ T /e/f ∪ T /e/g ∪ T − f − g /e ] obtained by flipping in e . Let B 1 and B 2 b e the extremities of P − e . W e no w do a case analysis on the t yp es of ˜ B 1 and ˜ B 2 . 3.2.1 ˜ B 1 or ˜ B 2 b elongs to F r ( G )[ T − f − g /e ] . Then F r ( G ) contains a Hamiltonian path that first visits F r ( G )[ T − e ] via P − e , then all the v ertices of F r ( G )[ T − f − g /e ] and finally those of F r ( G )[ T /e/f ∪ T /e/g ] (see Figure 12 ). B 1 ˜ B 1 P − e ⊂ F r ( G )[ T − e ] F r ( G )[ T − f − g /e ] F r ( G )[ T /e/f ∪ T /e/g ] Figure 12: Hamiltonian path if ˜ B 1 ∈ F r ( G )[ T − f − g /e ]. 3.2.2 ˜ B 1 or ˜ B 2 b elongs to F r ( G )[ T /e/f ∪ T /e/g ] \ { A ′ , A ′′ } . Without loss of generality , assume that ˜ B 1 b elongs to F r ( G )[ T /e/f ∪ T /e/g ] \ { A ′ , A ′′ } . By Theorem 3 , F r ( G )[ T /e/f ∪ T /e/g ] contains a Hamiltonian path starting at ˜ B 1 and ending at A ′ or A ′′ . As a result, F r ( G ) contains a Hamiltonian path visiting first P − e , then F r ( G )[ T /e/f ∪ T /e/g ] and finally F r ( G )[ T − f − g /e ] (see Figure 13 ). 3.2.3 ˜ B 1 and ˜ B 2 b elong to { A ′ , A ′′ } . Recall that G/e is of the form describ ed b y (4) of Theorem 16 b ecause A ′ = A ′ 1 and A ′′ = A ′′ 1 for all A ∈ T − f − g /e . Recall also that w is the v ertex of G/e resulting from the con traction of r and u . W e sa y that a vertex of G has depth i if it lies at distance i from w in G − f − g /e (that is at distance i from { r, u } 16 B 1 ˜ B 1 A ′ A ′′ P − e ⊂ F r ( G )[ T − e ] F r ( G )[ T − f − g /e ] F r ( G )[ T /e/f ∪ T /e/g ] Figure 13: Hamiltonian path if ˜ B 1 ∈ F r ( G )[ T /e/f ∪ T /e/g \ { A ′ , A ′′ } ]. in G − f − g ). W e label the vertices in V ( G ) \ { r, u, v } as in Theorem 16 : let v i b e the only vertex of V ( G − f − g /e ) \ { w , v } at depth i in the BFS starting from the ro ot w in G − f − g /e . Case 1: ˜ B 1  = ˜ B 2 . Without loss of generalit y , assume that ˜ B 1 = A ′ and ˜ B 2 = A ′′ . Claim 16.1. The outneighb ourho o d of r in G is c omp ose d of u which has depth 0, v 1 which has depth one, and v which has depth at le ast two. Pr o of of Claim. By definition, u is an outneigh b our of r and has depth 0 (and th us corresp onds to w in Figure 11 ). By (3) of Theorem 16 , the depth of v is at least tw o. On the other hand, other outneighbours of r in G are outneigh b ours of w in G − f − g /e and thus ha ve depth one, so { u, v } ⊆ N + ( r ) ⊆ { u, v 1 , v } . If v 1 is not an outneighbour of r in G , then r has outdegree 1 in G − e , so all arb orescences in T − e con tain the edge f = r v . Hence ˜ B 1 and ˜ B 2 b oth belong to T /e/f so ˜ B 1 = ˜ B 2 = A ′ whic h is a contradiction. ■ Assume that uv 1 / ∈ E ( G ). Then v 1 u ∈ E ( G ) b ecause the support of G − r is a clique. Recall that ( u, v ) was chosen to maximise N + ( u ) among all couples ( u, v ) ∈ N + ( r ) 2 with uv ∈ E ( G ), so | N + G ( u ) | ≥ | N + G ( v 1 ) | ≥ 2 because { u, v 2 } ⊆ N + G ( v 1 ). Hence, there exists x , an outneigh b our of u in G different from v . W e ha ve x ∈ N + G − f − g /e ( w ), hence x has depth one, that is x = v 1 , a contradiction. So uv 1 ∈ E ( G ) and b y Claim 16.1 , N + G ( u ) = { v 1 , v } . Th us v 1 is an outneigh- b our of b oth r and u . Consider the arb orescence A of G − f − g /e obtained b y p erforming a BFS staring at w . As v 1 has depth one, w v 1 ∈ E ( A ) and there are tw o arborescences A 1 and A 2 of G that are mapp ed to A when contracting e : one con tains the arcs r u and r v 1 , the other contains the arcs r u and uv 1 . So b oth of them con tain e but contain neither f nor g . Ho wev er, flipping f in them results in t w o differen t arborescences: as v  = v 1 , the arc uv 1 is con tained in A ′ 1 but not in A ′ 2 . This contradicts the fact that A ′ is constant ov er T − f − g /e . Hence w e cannot hav e ˜ B 1  = ˜ B 2 . 17 Case 2: ˜ B 1 = ˜ B 2 . Since ˜ B 1 = ˜ B 2 , the arb orescences B 1 and B 2 are a djacen t and P − e is in fact a Hamiltonian cycle. W e now proc eed similarly as after the deduction that F r ( G )[ T − f − g /e ] con tains a Hamiltonian cycle. Either ˜ B = ˜ B 1 for every arb orescence B in T − e or there are t wo arb orescences B 3 and B 4 , consecutiv e on P − e , suc h that ˜ B 3  = ˜ B 4 . In the second case, F r ( G )[ T − e ] con tains a Hamiltonian path whose extremities are B 3 and B 4 , so we are back in the case of one of the previous sections ( Section 3.2.2 or Section 3.2.3 ) or in the case of the previous paragraph. Th us we can assume that for ev ery B ∈ T − e , we hav e ˜ B = ˜ B 1 ∈ { A ′ , A ′′ } and th us that F r ( G )[ T − e ] is a clique. Theorem 16 can now be applied to G with w := r and v := u and to G/e with w := w and v := v to describe precisely the structure of G . Claim 16.2. Either G has a Hamiltonian p ath, or the outneighb ourho o d of r in G c onsists of u which has depth zer o, and v which has depth thr e e. Mor e over, al l arb or esc enc es of T − e c ontain f . Pr o of of Claim. W e proceed similarly as in Claim 16.1 . By (4) of Theorem 16 applied to G/e with w := w and v := v , w e hav e { u, v } ⊆ N + G ( r ) ⊆ N + G − f − g /e ( w ) ∪ { u } ⊆ { u, v , v 1 } and u has depth 0 while v has depth at least 2 (b y (3) ). If v has depth at least four, then by (4) of Theorem 16 applied to G/e with w := w and v := v , the graph G/e has t wo paths from the ro ot w to v 1 : w , v , v 1 and w , v , v 2 , v 3 , v 1 . Since v ∈ N + G ( r ), the graph G − e also contains tw o paths from the ro ot r to v 1 : r , v, v 1 and r , v , v 2 , v 3 , v 1 . By (2) of Theorem 16 applied to G with w := r and u := v , this contradicts the fact that ˜ B is constan t for B ∈ T − e . So the depth of v is either tw o or three. As f can alw a ys b e flipp ed in (b ecause it is inciden t to the ro ot) in G , one arb orescence of T − e con tains f , whic h implies that for ev ery B ∈ T − e , ˜ B = A ′ . So ev ery arbores cence of T − e con tains f , b ecause flipping in e results in A ′ . So v 1 is not an outneighbour of r , otherwise the path r, v 1 , . . . , v k − 1 , v can b e completed b y Theorem 2 into an arb orescence that con tains neither e nor f , a con tradiction, whic h pro ves that N + G ( r ) = { u, v } . W e can no w pro ve that v has depth exactly three. Assume tow ards con tra- diction that v has depth t wo. By Theorem 16 applied to G/e with w := w and v := v , the only p ossibilities for the subgraph of G induced b y { r , u, v , v 1 , v 2 } are those drawn on Figure 14 . F or an y of these p ossibilities, there are at most u v 1 v = v 2 r (a) u v 1 v 2 v r (b) Figure 14: The possible graphs G [ r, u, v , v 1 , v 2 ] if v has depth. The blac k arcs are con tained in G , while the red arcs are optional. t wo arb orescences of G [ r, u, v , v 1 , v 2 ] that contain the arcs e and f . W e also observ e from (4) that for every 3 ≤ i ≤ n , v i − 1 is the unique inneigh b our of v i 18 that is not one of its descendant. As a result, all arb orescences of G contain the path v 2 , . . . v 3 and T /e/f con tains at most tw o arborescences. This implies that T /e/f ∪ T /e/g is an edge or a 4-cycle. In either case, T contains a Hamiltonian path that first visits T − e , then T /e/f ∪ T /e/g and finally T − f − g /e . ■ W e first recall what we kno w ab out the outneighbourho o ds of r , u , v and v 1 . W e hav e N + ( r ) = { u, v } b y Claim 16.2 , N + ( u ) = { v 1 , v } b ecause v 1 has depth one and v 1 / ∈ N + ( r ). By (4) of Theorem 16 applied to G/e with w := w and v := v , w e hav e { v 1 } ⊆ N + ( v ) ⊆ { u, v 1 , v 2 } and N + ( v 1 ) = { v 2 } . So N + ( { r , u, v , v 1 } ) ∩ ( { v i : i ≥ 3 } \ { v } ) = ∅ and in ev ery arb orescence of T /e/g , there are only tw o p ossibilies for the path from r to v 2 : either r, u, v , v 2 or r , u, v , v 1 , v 2 . By (4) of Theorem 16 applied to G/e with w := w and v := v again, for all i ≥ 3, N − ( v i ) ∩ ( { r , u, v } ∪ { v j : j < i } ) = { v i − 1 } , a straightfor- w ard induction shows that in every arb orescence of T /e/g , there are only tw o p ossibilities for the path from r to v i : r , u, v , v 2 , . . . , v i or r, u, v , v 1 , v 2 , . . . , v i . As a result, there are at most t wo arb orescences in T /e/g . This implies that F r ( G )[ T /e/f ∪T /e/g ] is either an edge or a 4-cycle. In either case, there is a Hamil- tonian path F r ( G ), that first visits T − e , then T /e/f (b ecause by Claim 16.2 all arb orescences of T − e con tain f , so flipping in e gives A ′ ), then T /e/g and finally T − f − g /e . Ac kno wledgemen ts This w ork w as initiated during the workshop Order & Geometry 2024 (see https://sites.google.com/view/ordergeometry2024 ). 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