Graphs isomorphisms under edge-replacements and the family of amoebas
GGraphs isomorphisms under edge-replacementsand the family of amoebas
July 24, 2020
Yair Caro
Dept. of MathematicsUniversity of Haifa-OranimTivon 36006, [email protected]
Adriana Hansberg
Instituto de Matem´aticasUNAM JuriquillaQuer´etaro, [email protected]
Amanda Montejano
UMDI, Facultad de CienciasUNAM JuriquillaQuer´etaro, [email protected]
Abstract
Let G be a graph of order n and let e ∈ E ( G ) and e (cid:48) ∈ E ( G ) ∪ { e } . If the graph G − e + e (cid:48) is isomorphic to G , we say that e → e (cid:48) is a feasible edge-replacement . Wecall G a local amoeba if, for any two copies G , G of G that are embedded in K n , G can be transformed into G by a chain of feasible edge-replacements. On the otherhand, G is called global amoeba if there is an integer T ≥ G ∪ tK is a localamoeba for all t ≥ T . We study global and local amoebas under an underlying algebraictheoretical setting. In this way, a deeper understanding of their structure and theirintrinsic properties as well as how these two families relate with each other comes intolight. Moreover, it is shown that any connected graph can be a connected componentof an amoeba, and a construction of a family of amoeba trees with a Fibonacci-likestructure and with arbitrary large maximum degree is presented. Graphs called amoebas first appeared in [3] where certain Ramsey-Tur´an extremal problemswere considered, which dealt with the existence of a given graph with a prescribed colorpattern in 2-edge-colorings of the complete graph. More precisely, amoebas arose fromthe search of a graph family with certain interpolation properties that are suitable for thetechniques to show balanceability or omnitonal properties, see [2, 3] for a deeper insightinto this matter. For the interested reader, we refer to [1, 6, 7, 9, 10, 12, 14] for moreliterature related to interpolation techniques in graphs. The feature that makes amoebas1 a r X i v : . [ m a t h . C O ] J u l ork are one-by-one replacements of edges, where, at each step, some edge is substitutedby another such that an isomorphic copy of the graph is created. Similar edge-operationshave been studied, for instance, in [4, 5, 8, 11, 13]. We note here that, in this paper, we willdistinguish between two different graph classes which we will call local or global amoebas.The amoebas defined in [3] correspond to the class of global amoebas.Suppose we have a graph G embedded in K n , where n is large. A global amoeba is,informally, a graph G that can be transformed into any other embedded copy of it by a seriesof substitutions of an edge by another, preserving in each step the structure of the graph,i.e., such that we obtain an isomorphic copy of the previous graph. For example, it can beeasily checked that the path P k on k ≥ P = v v . . . v k is embedded in K n . Then we can remove the edge v k − v k from P and include the edge v v k , so that the new graph is again a path on k vertices. Similarly, we can take any vertex v ∈ V ( K n ) \ V ( P ) and substitute the edge v k − v k with the edge v k − v . With these twooperation-types, we can obtain a series of paths whose last member is certain given copy P (cid:48) of P . Clearly, such a chain of operations can happen if n is large enough and it is not clearat a first look how large the n needs to be at least. Interestingly, it turns out that we justneed n to have one unit larger than the order of P , and that occurs for any global amoeba.This is shown in Theorem 3.7, which is a major achievement of this paper. A family ofgraphs that is closely related to global amoebas are those which we call local amoebas andtheir definition is the same as for global amoebas with the only difference that they areassumed to be embedded in K n ( G ) . It is easy to see that paths are local amoebas, too.However, for example, a complete graph minus an edge is a local amoeba but not a globalamoeba. We will give these and other more detailed examples further on, which will also beshown formally to which family they do or do not belong. The direct connection betweenthese two graph families can be already seen in their formal definition (Definition 3.1), thatwill be given further on.A first encounter with amoebas gives the impression that such graphs are very rareand have a very simple structure. This, however, is not the case and amoebas may havequite a complicated structure. In this paper, we consider amoebas in detail focusing onseveral methods to build and to characterize them. In order to formalize all concepts andset definitions, we need first an algebraic theoretical setting. This approach may appearunnecessarily complicated on a first glance, but it will reveal its power soon.The paper is organized as follows. In Section 2, we present the group theoretical back-ground with which we will be able to model how the operations that we will be performingon a graph G , the so-called feasible edge-replacements, work such that always isomorphiccopies of G are obtained. With the help of this algebraic setting, we will formally intro-duce, in Section 3, the concepts of global and local amoeba, and we will demonstrate severalstructural properties of both graph families, establishing also very clearly the relation be-tween them as well as their differences. In order to show the purpose of the results, we willillustrate with abundant examples. Finally, in Section 4, we will present some interestingconstructions of global amoebas. In particular, we will show that any connected graph canbe a connected component of a global amoeba, from which follows that, for example, wecan have global amoebas with arbitrary large clique or chromatic number, and we will con-struct a family of global amoeba-trees with a Fibonacci-like structure and with arbitrarylarge maximum degree. 2 Theoretical setting
Let [ n ] = { , , ..., n } and let S n be the symmetric group, whose elements are permutations of[ n ]. The group of automorphisms of a graph G is denoted by Aut( G ). Thus, Aut( K n ) ∼ = S n where K n is the complete graph of order n and, for any graph G of order n , Aut( G ) ∼ = S forsome S ≤ S n . Let V = { v , v , . . . , v n } be the set of vertices of K n . Let G be a spanningsubgraph of K n defined by its edge set E ( G ) ⊆ E ( K n ) and let L G = { ij | v i v j ∈ E ( G ) } ,where we do not distinguish between ij and ji . For each σ ∈ S n , we define λ σ : V → [ n ] asthe labeling of the vertices of K n defined by λ σ ( v i ) = σ ( i ) and consider the copy G σ of G embedded in K n defined by E ( G σ ) = { v σ − ( i ) v σ − ( j ) | ij ∈ L G } . Hence, each labeled copy of G embedded in K n correspond to a permutation σ ∈ S n and viseversa. Observe that, for every (non-labeled) subgraph G (cid:48) of K n isomorphic to G there are | Aut( G ) | different labelings of V that correspond to G (cid:48) , that is, the set { σ ∈ S n | G σ = G (cid:48) } has | Aut( G ) | elements. Moreover, { σ ∈ S n | G σ = G } ∼ = Aut( G ). We will set A G = { σ ∈ S n | G σ = G } . Example 2.1.
Let G = P with V ( P ) = { v , v , v , v } and E ( P ) = { v v , v v , v v } .Thus, L G = { , , } and { σ ∈ S | G σ = G } = { id, (14)(23) } ∼ = Aut( P ) . For G (cid:48) , theisomorphic copy of G defined by E ( G (cid:48) ) = { v v , v v , v v } , we have two permutations,namely (23) and (14) , that satisfy G (23) = G (14) = G (cid:48) . See Figure 1 to visualize thecorresponding labelings and observe that, in all cases, E ( G σ ) = { v σ − ( i ) v σ − ( j ) | ij ∈ L G } .For example, if σ = (23) then E ( G σ ) = { v σ − ( i ) v σ − ( j ) | ij ∈ L G } = { v v , v v , v v } . v
For G = P with V ( P ) = { v , v , v , v } and E ( P ) = { v v , v v , v v } , we have L G = { , , } . The labelings corresponding to the permutations id, (14)(23) , (23) and (14) aredepicted (left to right) showing the copies G = G id = G (14)(23) and G (cid:48) = G (23) = G (14) , where E ( G σ ) = { v σ − ( i ) v σ − ( j ) | ij ∈ L G } in all cases. It is important to note that the set of labels L G σ = { σ ( i ) σ ( j ) | v i v j ∈ E ( G σ ) } of theedges of G σ is the same for all σ ∈ S n , i.e. L G σ = L G for all σ ∈ S n . Moreover, thecorresponding copies of the vertices and edges of G in G σ are given by their labels: the copyof vertex v i of G is the vertex of G σ having label i , while the copy of an edge v i v j ∈ E ( G )is the edge of G σ having label ij . 3iven e ∈ E ( G ) and e (cid:48) ∈ E ( G ) ∪ { e } , we say that the graph G − e + e (cid:48) is obtained from G by performing the edge-replacement e → e (cid:48) . If G − e + e (cid:48) is a graph isomorphic to G , wesay that the edge-replacement e → e (cid:48) is feasible . Let R G = { rs → kl | G − v r v s + v k v l ∼ = G } be the set of all feasible edge-replacements of G given by their labels. Notice that, sincefeasible edge-replacements are defined by the labels of the edges, any rs → kl ∈ R G repre-sents also a feasible edge-replacement of any copy G ρ , ρ ∈ S n . Hence, clearly R G ρ = R G for any ρ ∈ S n .Given a feasible edge-replacement, rs → kl ∈ R G , we will use the following notation S G ( rs → kl ) = { σ ∈ S n | G σ = G − v r v s + v k v l } . We will use sometimes the notation e → e (cid:48) ∈ R G when we do not require to specify theindexes of the vertices involved in the edge-replacement.Now we can state the following lemma that will establish the ground for how we aregoing to work with the very interesting graph family of the amoebas. We use right to leftnotation for the composition of permutations, that is, σ ◦ ρ ∈ S n is defined as σ ( ρ ( x )) forevery x ∈ [ n ]. We omit the symbol “ ◦ ” when there is no confusion. Lemma 2.2.
Let G be a graph defined on the vertex set V = { v , v , . . . , v n } and let L G = { ij | v i v j ∈ E ( G ) } . For any rs → kl ∈ R G , σ ∈ S G ( rs → kl ) and ρ ∈ S n , we havethe following:(i) E ( G σ ) = { v i v j | ij ∈ ( L G \ { rs } ) ∪ { kl }} . (ii) ( L G \ { rs } ) ∪ { kl } = (cid:8) σ − ( i ) σ − ( j ) | ij ∈ L G (cid:9) . (iii) G σ ρ = G ρ − e + e (cid:48) , where e = v ρ − ( r ) v ρ − ( s ) and e (cid:48) = v ρ − ( k ) v ρ − ( l ) .Proof. (i) Since σ ∈ S G ( rs → kl ), by definition, we have E ( G σ ) = ( E ( G ) \ { v r v s } ) ∪ { v k v l } = ( { v i v j | ij ∈ L G } \ { v r v s } ) ∪ { v k v l } = { v i v j | ij ∈ ( L G \ { rs } ) ∪ { kl }} . (ii) By (i) and definition of G σ , we have { v i v j | ij ∈ ( L G \ { rs } ) ∪ { kl }} = E ( G σ ) = (cid:8) v σ − ( i ) v σ − ( j ) | ij ∈ L G (cid:9) , from which, by taking the set of pairs of sub-indexes, we obtain( L G \ { rs } ) ∪ { kl } = (cid:8) σ − ( i ) σ − ( j ) | ij ∈ L G (cid:9) . (iii) We need to prove that the copy of G associated to the composition σ ρ can be ob-tained by applying the edge-replacement e → e (cid:48) to G ρ , where e = v ρ − ( r ) v ρ − ( s ) and4 (cid:48) = v ρ − ( k ) v ρ − ( l ) . Observe that in G ρ the edge e = v ρ − ( r ) v ρ − ( s ) is labeled with rs while the edge e (cid:48) = v ρ − ( k ) v ρ − ( l ) is labeled with kl . Then with (ii), we deduce E ( G ρ − e + e (cid:48) ) = (cid:8) v ρ − ( i ) v ρ − ( j ) (cid:12)(cid:12) ij ∈ ( L G \ { rs } ) ∪ { kl } (cid:9) = (cid:8) v ρ − ( σ − ( i )) v ρ − ( σ − ( j )) | ij ∈ L G (cid:9) = E ( G σ ρ ) . To the sake of comprehension, we show a concrete example.
Example 2.3.
Let G = P with V ( P ) = { v , v , v , v } and E ( P ) = { v v , v v , v v } .Then, R G = { → , → , → , → , → , → , → , → } . Consider → ∈ R G which corresponds to the feasible edge-replacement v v → v v (seeFigure 2). Observe that the graph G − v v + v v corresponds to G σ for all σ ∈ S G (12 →
14) = { (24) , (1432) } . Set σ = (24) . Now we formulate the three items of Lemma 2.2 forthis example. Recall that L = { , , } and so ( L \ { } ) ∪ { } = { , , } . Also notethat σ − = (24) .(i) E ( G σ ) = { v i v j | ij ∈ ( L \ { } ) ∪ { }} = { v v , v v , v v } . (ii) By (i) and definition of G σ , we have { v v , v v , v v } = E ( G σ ) = (cid:8) v σ − ( i ) v σ − ( j ) | ij ∈ L (cid:9) = { v v , v v , v v } , thus the sets of pairs of sub-indices coincide ( L \ { } ) ∪ { } = { , , } = { , , } = (cid:8) σ − ( i ) σ − ( j ) | ij ∈ L (cid:9) . (iii) To illustrate this item we consider ρ = (23) , and so ρ − = (23) . We need to showthat the copy of G associated to the composition σ ρ = (24) ◦ (23) = (234) is obtainedby applying the edge-replacement e → e (cid:48) to G ρ , where e = v ρ − (1) v ρ − (2) = v v and e (cid:48) = v ρ − (1) v ρ − (4) = v v . Note that, in G ρ , the edge e is labeled with while theedge e (cid:48) is labeled with and see Figure 3. G = G id
With G = P defined as in Figure 1, we perform the feasible edge replacement 12 → G σ of G where σ = (24). !
With G = P defined as in Figure 1, we perform the feasible edge-replacement given by12 → ∈ R G in the copy G ρ of G where ρ = (23), obtaining the copy G σ ρ = G (234) . Item (iii) of Lemma 2.2 means that performing a feasible edge-replacement e → e (cid:48) ∈ R G in a copy G ρ of G yields the copy of G given by the permutation σ ρ , where we can chooseany σ ∈ S G ( e → e (cid:48) ). It now makes sense to consider the group S G generated by thepermutations associated to all feasible edge-replacements, that is, by the set E G = (cid:91) e → e (cid:48) ∈ R G S G ( e → e (cid:48) ) . Thus, S G = (cid:104)E G (cid:105) . Clearly, S G acts on the set { G ρ | ρ ∈ S n } by means of ( σ, G ρ ) (cid:55)→ G σρ , where σ ∈ S G and ρ ∈ S n . Observe that this action represents what happens when a series of feasibleedge-replacements, represented by σ , is performed on a copy G ρ of G : the result is thegraph G σρ . We shall also note that a trivial edge-replacement , i.e. an edge-replacement rs → kl where { r, s } = { k, l } , is always feasible, and Aut( G ) ∼ = { σ ∈ S n | G σ = G } ≤ S G .The following observation is straightforward from the definition of feasible edge-replacementand item ( iii ) of Lemma 2.2. Observation 2.4.
Let G be a graph defined on the vertex set V = { v , v , . . . , v n } . For any v i ∈ V , rs → kl ∈ R G , σ ∈ S G ( rs → kl ) and ρ ∈ S n , we have deg G σρ ( v i ) = deg G ρ ( v i ) − , if i ∈ { r, s } \ { k, l } deg G ρ ( v i ) + 1 , if i ∈ { k, l } \ { r, s } deg G ρ ( v i ) , else. In the next lemma, we discuss the connection between the feasible edge-replacementsof a graph G and those of its complement graph G , concluding that the correspondingassociated groups are the same. Lemma 2.5.
Let G be a graph defined on the vertex set V = { v , v , . . . , v n } . Then,(i) For any σ ∈ S G , G σ = G σ .(ii) rs → kl ∈ R G if and only if kl → rs ∈ R G . iii) S G ( rs → kl ) = S G ( rs → kl ) .(iv) S G = S G .Proof. (i) The statement follows from, E ( G σ ) = E ( K n ) \ { v σ − ( i ) v σ − ( j ) | ij ∈ L } = { v σ − ( i ) v σ − ( j ) | ij / ∈ L } = E ( G σ ) . (ii) Let rs → kl ∈ R G and σ ∈ S G ( rs → kl ). Then G σ = G − v r v s + v k v l = G − v k v l + v r v s .Since G σ ∼ = G , then G σ ∼ = G , and thus we deduce that G − v k v l + v r v s ∼ = G , implying that kl → rs ∈ R G . The converse is analogous.(iii) By what we showed in items (i) and (ii), we have G σ = G σ = G − v k v l + v r v s = G τ forany τ ∈ S G ( kl → rs ). It follows that σ ∈ S G ( kl → rs ). Hence, S G ( rs → kl ) ⊆ S G ( kl → rs ).The other inclusion is analogous.(iv) By items (ii) and (iii), we have S G = (cid:104) σ | σ ∈ S G ( rs → kl ) for some rs → kl ∈ R G (cid:105) = (cid:104) σ | σ ∈ S G ( kl → rs ) for some kl → rs ∈ R G (cid:105) = S G . We shall note that the graphs that we have considered are not necessarily connected.Actually, we will work with (non-connected) graphs containing isolated vertices. To finishthis section, we establish important facts related to the feasible edge-replacements in suchgraphs. For a group S ≤ S n acting on [ n ] and a subset X ⊆ [ n ], we denote by Stab S ( X ),the stabilizer of S on X , that isStab S ( X ) = { σ ∈ S | σ ( x ) ∈ X for all x ∈ X } . Lemma 2.6.
Let G = H ∪ H (cid:48) be the disjoint union of two graphs H and H (cid:48) , where G has order n and H has order m < n . Let V ( H ) = { v , v , . . . , v m } and V ( H (cid:48) ) = { v m +1 , v m +2 , . . . , v n } . For each σ ∈ S H and τ ∈ R H (cid:48) , define ρ = ρ ( σ, τ ) ∈ S n as ρ ( i ) = (cid:26) σ ( i ) , if i ∈ [ m ] τ ( i ) , else,Then R H , R H (cid:48) ⊆ R G and S H × S H (cid:48) ∼ = { ρ ( σ, τ ) | σ ∈ S H , τ ∈ R H (cid:48) } ≤ Stab S G ([ m ]) .Proof. That R H , R H (cid:48) ⊆ R G is easy to see. Let σ ∈ S H and τ ∈ R H (cid:48) . By definition, σ = σ q σ q − · · · σ for certain σ , · · · , σ q ∈ E H , while τ = τ q (cid:48) τ q (cid:48) − · · · τ for certain τ , · · · , τ q (cid:48) ∈E H (cid:48) . Without loss of generality, assume that q ≥ q (cid:48) . Define τ j = id S H (cid:48) for q (cid:48) + 1 ≤ j ≤ q .For 1 ≤ j ≤ q , let ρ j ( i ) = (cid:26) σ j ( i ) , if i ∈ [ m ] τ j ( i ) , else.Since R H , R (cid:48) H ⊆ R G and, for 1 ≤ i ≤ q , G ρ j = ( H ∪ H (cid:48) ) ρ j = H σ j ∪ H (cid:48) τ j , then ρ , · · · , ρ q ∈ E G . Moreover, ρ = ρ q ρ q − · · · ρ and ρ ∈ Stab S G ([ m ]). Since, clearly S H × S H (cid:48) ∼ = { ρ ( σ, τ ) | σ ∈ S H , τ ∈ R H (cid:48) } , the latter is a subgroup of Stab S G ([ m ]).7 emark 2.7. In view of Lemma 2.6, we will identify the groups { ρ ( σ, τ ) | σ ∈ S H , τ ∈ R H (cid:48) } and S H × S H (cid:48) and we will use the notation ( σ, τ ) instead of ρ ( σ, τ ) . Since we have that S H × S H (cid:48) ≤ Stab S G ([ m ]) ≤ S G , we have in particular that S H ∼ = S H × (cid:104) id S H (cid:48) (cid:105) ≤ S G andthat S H (cid:48) ∼ = (cid:104) id S H (cid:105) × S H (cid:48) ≤ S G . Hence, again in an abuse of notation, we will say that S H and S H (cid:48) are subgroups of S G . Lemma 2.8.
Let G be a graph and G ∗ = G ∪ tK for some t ≥ . Let V ( G ) = { v , v , . . . , v n } and V ( G ∗ ) = { v , v , . . . , v n + t } . Then we have the following properties.(i) For any e → e (cid:48) ∈ R G and σ ∈ S G ∗ ( e → e (cid:48) ) , the permutation (cid:98) σ ∈ S n defined by (cid:98) σ ( i ) = σ ( i ) for i ∈ [ n ] satisfies (cid:98) σ ∈ S G ( e → e (cid:48) ) .(ii) If e → e (cid:48) ∈ R G ∗ \ R G , then e (cid:54) = e (cid:48) and we have one of two cases. • e = rs and e (cid:48) = kl for some r, s, k ∈ [ n ] and l ∈ [ n + 1 , n + t ] such that deg G ∗ ( v r ) = 1 . Moreover, S G ∗ ( e → e (cid:48) ) = { ϕ ◦ ( r l ) | ϕ ∈ A G ∗ } . • e = rs and e (cid:48) = kl for some r, s ∈ [ n ] and k, l ∈ [ n + 1 , n + t ] such that deg G ∗ ( v r ) = deg G ∗ ( v s ) = 1 . Moreover, S G ∗ ( e → e (cid:48) ) = { ϕ ◦ ( r k )( s l ) | ϕ ∈ A G ∗ } .Proof. (i) Let e → e (cid:48) ∈ R G and σ ∈ S G ∗ ( e → e (cid:48) ), where e = rs and e (cid:48) = kl , for certain r, s, k, l ∈ [ n ]. Then G ∗ σ = G ∗ − v r v s + v k v l = ( G − v r v s + v k v l ) ∪ tK = G (cid:98) σ ∪ tK . Hence, (cid:98) σ ∈ S G ( e → e (cid:48) ).(ii) Any feasible edge-replacement e → e (cid:48) ∈ R G ∗ \ R G , where e = rs and e (cid:48) = kl is suchthat it involves an edge v r v s in G and an edge in v k v l ∈ E ( G ∗ ) \ E ( G ). Hence, at leastone of v k , v l has degree 0, say deg G ∗ ( v l ) = 0. Since the degree sequence is preserved afterthe edge-replacement, we have that at least one of v r , v s is of degree 1, say deg G ∗ ( v r ) = 1.Suppose first that k ∈ [ n ]. Then r, s, k ∈ [ n ] and l ∈ [ n + 1 , n + t ] such that deg G ∗ ( v r ) = 1.It is also easy to see that { ϕ ◦ ( r l ) | ϕ ∈ A G ∗ } ⊆ S G ∗ ( e → e (cid:48) ). On the other hand, for σ ∈ S G ∗ ( e → e (cid:48) ), we know that E ( G ∗ σ ) = E (cid:16) G ∗ ( r l ) (cid:17) , from which we can deduce that thereis a ϕ ∈ A G ∗ such that G ∗ σ = (cid:16) G ∗ ( r l ) (cid:17) ϕ = G ∗ ϕ ◦ ( r l ) . From the latter we obtain that σ = ϕ ◦ ( r l ), and thus S G ∗ ( e → e (cid:48) ) ⊆ { ϕ ◦ ( r l ) | ϕ ∈ A G ∗ } . Altogether, we have S G ∗ ( e → e (cid:48) ) = { ϕ ◦ ( r l ) | ϕ ∈ A G ∗ } . On the other hand,if k ∈ [ n + 1 , n + t ], then we have r, s ∈ [ n ] and k, l ∈ [ n + 1 , n + t ], from which wededuce that v s has degree 1, too. Hence, deg G ∗ ( v r ) = deg G ∗ ( v s ) = 1. It is also clear that ϕ ◦ ( r k )( s l ) ∈ S G ∗ ( e → e (cid:48) ) for any ϕ ∈ A G ∗ . To show the other contention direction, weproceed analogously to previous case. In the previous section we define, for any (not necessarily connected) graph G of order n ,a subgroup S G of S n generated by the set of permutations in S n associated to the feasibleedge-replacements of G . By means of this group, we are ready to define both types ofamoebas. 8 efinition 3.1. A graph G of order n is called a local amoeba if S G = S n . That is,any labeled copy of G embedded in K n can be reached, from G , by a chain of feasible edge-replacements. On the other hand, a graph G is called global amoeba if there is an integer T ≥ such that G ∪ tK is a local amoeba for all t ≥ T . Note that for the concept of global amoeba, which is the one considered already in theliterature [3], we can maintain the image of a graph G embedded in a complete graph K N ,with N = n + t much larger than | V ( G ) | = n , traveling via feasible edge replacements fromany given copy of it to any other one.It is not difficult to convince oneself that, for every n ≥
2, a path P n is both a localamoeba and a global amoeba, while a cycle C n is neither a local amoeba nor a globalamoeba, for any n ≥
3. After developing some theory, we will provide formal argumentsto prove the above facts and, also, to prove rigorously all statements in the next example,in which we exhibit interesting graphs and families of graphs concerning all possibilities ofbeing, or not, a local or a global amoeba.
Example 3.2.
1. The following graphs are neither local nor global amoebas:(a) The star K ,k − on k vertices, for k ≥ .(b) Every (non-complete) r -regular graph, for r ≥ .2. The following graphs are are both, local amoeba and global amoebas:(a) The path P k on k vertices, for k ≥ .(b) The graph C ( k, obtained from a cycle on k vertices by attaching a pendantvertex, for k ≥ .3. The following graphs are are local but not global amoebas:(a) The graph K − k obtained from the complete graph K k by deleting one edge, for k ≥ .(b) The graph C +5 obtained from a cycle on five vertices by adding one edge betweentwo diametrical vertices.4. The following graphs are global but not local amoebas:(a) tP k , for t ≥ and k ≥ .(b) The graph G of order depicted in Figure 4. v
A connected graph which is global amoeba but not local amoeba.
9e point out that some of the statements in Example 3.2 are easy to prove but someothers are not. For instance, consider the graph depicted in Figure 4: at a first glance, itis not clear why this graph is a global amoeba but not a local amoeba. In the followingsection, we will give structural results that will help us understand the families of local andglobal amoebas, as well as the relationship between them.
We begin by noticing simple properties.
Proposition 3.3.
Let G be a graph of order n .(i) G is a local amoeba if and only if G is a local amoeba.(ii) If all feasible edge-replacements of G are trivial, then G is a local amoeba if and onlyif G = K n or G = K n .Proof. Item (i) follows from item (iv) of Lemma 2.5 and the definition of local amoeba.To prove item (ii), let G be a graph with only trivial feasible edge-replacements, then E G = { σ ∈ S n | G σ = G } ∼ = Aut( G ). Since, by definition, S G = (cid:104)E G (cid:105) , then S G = S n if andonly if Aut( G ) ∼ = S n , which holds precisely when G = K n or G = K n .Next we prove a very useful fact concerning the degree sequences of local and globalamoebas. Proposition 3.4.
Let G be a graph of order n with minimum degree δ and maximumdegree ∆ . If G is a local amoeba then, for every integer r with δ ≤ r ≤ ∆ , there is avertex v ∈ V ( G ) with deg G ( v ) = r . If G is a global amoeba the same is true and, moreover, δ ∈ { , } .Proof. Let G be a local amoeba with V = { v , v , . . . , v n } and let k, l ∈ [ n ] be such thatdeg G ( v k ) = δ and deg G ( v l ) = ∆. Since G is a local amoeba, S G ∼ = S n , implying that thereis a τ ∈ S G such that τ = ( k l ). Let σ , · · · , σ q ∈ E G be such that τ = σ q σ q − · · · σ . Nowset τ i = σ i σ i − · · · σ , for 1 ≤ i ≤ q , and τ = id. In particular, we have τ q = τ . Considernow the sequence (cid:16) deg G τi ( v k ) (cid:17) ≤ i ≤ q . We know the first and the last values of this sequence, namely deg G τ ( v k ) = deg G ( v k ) = δ and deg G τq ( v k ) = deg G τ ( v k ) = deg G ( kl ) ( v k ) = deg G ( v l ) = ∆. If r ∈ { δ, ∆ } , we are done.Now, suppose there is an integer r with δ < r < ∆ such that deg G τi ( v k ) (cid:54) = r for all 0 ≤ i ≤ q .Let j be the first index where deg G τj ( v k ) ≥ r + 1. Then we have deg G τj − ( v k ) ≤ r −
1. But,since G τ j is obtained by performing a feasible edge-replacement in G τ j − , by Observation 2.4,we have | deg G τj ( v k ) − deg G τj − ( v k ) | ≤
1, which is not possible in this case. Hence, we obtaina contradiction and it follows that there is a j such that deg G τj ( v k ) = r . Since G τ j ∼ = G , itfollows that G has a vertex of degree r .If G is a global amoeba, by definition, G ∪ tK is a local amoeba for some t ≥
1. Bythe previous case, this local amoeba satisfies the desired degree sequence condition and,moreover, necessarily δ ∈ { , } . 10he next observation will be a useful tool to determine if a graph G is a local amoeba.To continue we need some terminology. Given a subgroup S ≤ S n and k ∈ [ n ], we denoteby Sk the orbit of k by means of the canonical action of S on [ n ], i.e. Sk = { σ ( k ) | σ ∈ S } . Also, we use Stab S ( k ) = { σ ∈ S | σ ( k ) = k } . By the well known fact that the symmetricgroup S n is generated by (cid:104) S ∪ { ( i k ) }(cid:105) , where S is a transitive subgroup of Stab S n ( k ) forsome k ∈ [ n ], we have the following observation. Observation 3.5.
Let G be a graph of order n . Then G is a local amoeba if and only ifthere is a k ∈ [ n ] such that Stab S G ( k ) acts transitively on [ n ] \ { k } and ( j k ) ∈ S G for some j ∈ [ n ] \ { k } . By means of Observation 3.5, we get the following.
Lemma 3.6. If G is a local amoeba with δ ( G ) ∈ { , } , then G ∪ K is a local amoeba.Proof. Let G be a local amoeba defined on the vertex set V = { v , v , . . . , v n } . If G = K n , weare done by Proposition 3.3 (ii). Hence, in view of Proposition 3.4 and because δ ( G ) ∈ { , } ,we can assume that G has a vertex of degree 1, say deg G ( v n ) = 1. Consider now the graph G ∪ K defined on the vertex set V ∪ { v n +1 } . Consider, as in Lemma 2.6, the permutations( σ, id) ∈ S G ∪ K , where σ ∈ S G . Moreover, S n ∼ = S G ∼ = (cid:104) ( σ, id) | σ ∈ S G (cid:105) ≤ Stab S G ∪ K ( n + 1)which acts transitively on [ n ]. Also note that ( n n + 1) ∈ S G ∪ K (by means of the feasibleedge-replacement rn → r ( n + 1) where v r is the only neighbor of v n ). With the useof Observation 3.5, we conclude that S G ∪ K = S n +1 , implying that G ∪ K is a localamoeba.Now we are ready to prove a theorem that gives equivalent statements for the definitionof a global amoeba. Theorem 3.7.
Let G be a graph defined on the vertex set V = { v , v , . . . , v n } . Thefollowing statements are equivalent:(i) G is a global amoeba.(ii) For each x ∈ [ n ] , there is a y ∈ S G x such that deg G ( v y ) = 1 .(iii) G ∪ K is a local amoeba.(iv) G ∪ tK is a local amoeba for some t ≥ .Proof. We will show (i) ⇒ (ii) ⇒ (iii) ⇒ (iv) ⇒ (i). To see (i) ⇒ (ii), let G be a globalamoeba. By definition, we know that for some t ≥ G ∗ = G ∪ tK is a local amoeba,that is, S G ∗ ∼ = S n + t . Let V ( G ∗ ) = V ∪ { v n +1 , ..., v n + t } and let x ∈ [ n ]. Take a permutation τ ∈ S G ∗ with τ ( x ) = l for some l ∈ [ n + 1 , n + t ]. We know that τ = σ q σ q − · · · σ where σ , · · · , σ q ∈ E G ∗ . Set τ i = σ i . . . σ , 1 ≤ i ≤ q . We will show that τ can been chosen havingthe following properties: 11a) τ i ( x ) ∈ [ n ] for all 1 ≤ i ≤ q − τ i ( x ) (cid:54) = τ j ( x ) for any pair i, j with 1 ≤ i < j ≤ q .(c) σ i ∈ Stab([ n ]), for 1 ≤ i ≤ q − τ i ( x ) ∈ [ n + 1 , n + t ] for some i < q , then we can take τ i instead of τ . Hence, wemay assume property (a). If τ i ( x ) = τ j ( x ) for some pair 1 ≤ i < j ≤ q , then we cantake τ (cid:48) = σ q σ q − . . . σ j +1 σ i σ i − . . . σ in stead of τ . Hence, we may assume (b). Suppose σ j / ∈ Stab([ n ]) for some j ∈ { , , . . . , q − } . Choose j such that it is minimum withthis property. Then σ j ( r ) = l for some r ∈ [ n ] \ { τ j − ( x ) } and some l ∈ [ n + 1 , n + t ].By Lemma 2.8 (ii), either σ j = ϕ ◦ ( r l ) for some ϕ ∈ A G ∗ , or σ j = ϕ ◦ ( r l )( s k ) forsome s ∈ [ n ] \ { r, τ j − ( r ) } , k ∈ [ n + 1 , n + t ] \ { l } , and for some ϕ ∈ A G ∗ . Suppose wehave the first, i.e., σ j = ϕ ◦ ( r l ). Since { ( r l ) , ϕ } ⊆ E G ∗ , ( r l )( τ j − ( m )) = τ j − ( m ), and ϕ ( τ j − ( x )) = τ j ( x ), we can replace τ by τ (cid:48) = σ (cid:48) q . . . σ (cid:48) , with σ (cid:48) i = σ i for 1 ≤ i ≤ q , i (cid:54) = j ,and σ (cid:48) j = ϕ . Thus, we can assume that σ i ∈ Stab([ n ]), for 1 ≤ i ≤ q − σ j = ϕ ◦ ( r l )( s k ) is completely analogous. Now, since τ q − ( x ) ∈ [ n ] and σ q ( τ q − ( x )) = τ q ( x ) = τ ( x ) = l ∈ [ n + 1 , n + t ], it follows by Lemma 2.8(ii), that σ q = ρ ◦ ( l y ) for some ρ ∈ A G ∗ and y = τ q − ( x ), where deg G ∗ ( v y ) = 1. Finally,we will show that y ∈ S G x . To this aim, for each 1 ≤ i ≤ q −
1, we define a permutation (cid:98) σ i ∈ S n by (cid:98) σ i ( j ) = σ i ( j ) for all j ∈ [ n ], which by Lemma 2.8 (i) satisfies (cid:98) σ i ∈ S G . Then( (cid:98) σ q − . . . (cid:98) σ )( x ) = ( σ q − . . . σ )( x ) = τ q − ( x ) = y, implying that y ∈ S G x . Since deg G ( v y ) = deg G ∗ ( v y ) = 1, we have finished.To see (ii) ⇒ (iii), let G ∗ = G ∪ K with V ( G ∗ ) = V ∪ { v n +1 } . Assume, without loss ofgenerality, that v , . . . , v p are the vertices of degree 1 in G . Note that item (iii) is equivalentto say that [ n ] = ∪ pi =1 S G i . For every σ ∈ S G , consider, as in Lemma 2.6, the permutations( σ, id) ∈ Stab S G ∗ ( n + 1). Let k ∈ S G i for some i ∈ [ p ]. Then there is a σ ∈ S G such that σ ( k ) = i . Moreover ( i n + 1) ∈ S G ∗ for every i ∈ [ p ] because of the feasible edge-replacement s i i → s i ( n + 1) ∈ R G ∗ , where v s i is the unique neighbor if v i in G . Then( σ, id) − ( i n + 1)( σ, id)( j ) = n + 1 , if j = kk, if j = n + 1 j, else.Hence, ( σ, id) − ( i n + 1)( σ, id) = ( k n + 1) ∈ S G ∗ . Since this holds for each k ∈ (cid:83) pi =1 S G i =[ n ], ( k n + 1) ∈ S G ∗ for all k ∈ [ n ] and we conclude that S G ∗ ∼ = S n +1 . Hence, G ∗ = G ∪ K is a local amoeba.The implication (iii) ⇒ (iv) is direct and the fact that item (iv) implies item (i) followsby Lemma 3.6 and the definition of global amoeba.Observe that from the proof of implication (i) ⇒ (ii), in view of Observation 2.4, wecan deduce that actually, a graph G is a global amoeba if and only if for each x ∈ [ n ] suchthat deg G ( v x ) ≥
2, there is a σ ∈ S G such that deg G ( v σ ( x ) ) = deg G ( v x ) − orollary 3.8. Let H and H (cid:48) be two vertex-disjoint global amoebas. Then G = H ∪ H (cid:48) isa global amoeba, too.Proof. Let V ( H ) = { v , v , . . . , v m } and V ( H (cid:48) ) = { v m +1 , v m +2 , . . . , v n } . Let I H and I H (cid:48) bethe sets of all indexes of the vertices of degree one in H and H (cid:48) , respectively. Since H and H (cid:48) are global amoebas, we have, by the equivalence of items (i) and (ii) of Theorem 3.7,that (cid:91) i ∈ I H S H i = [ m ] and (cid:91) i ∈ I H (cid:48) S H (cid:48) i = [ m + 1 , n ] . Hence, with S ≤ S G the soubgroup isomorphic to S H × S H (cid:48) (see Lemma 2.6) and I = I H ∪ I H (cid:48) , we obtain [ n ] = (cid:91) i ∈ I Si ⊆ (cid:91) i ∈ I S G i, from which, again by the equivalence of items (i) and (ii) of Theorem 3.7, we obtain that G is a global amoeba.Observe that the converse statement of Corollary 3.8 is not valid. For example, let H = P k and H (cid:48) = C k , for k ≥
3. The graph G = H ∪ H (cid:48) is a global amoeba (see item 4 (b)of Example 3.2). However, H = C k is not a global amoeba (see item 1 (a) of Example 3.2).We remark also at this point that there is no corresponding result to Corollary 3.8 forlocal amoebas, since the union of two local amoebas is not necessarily again a local amoeba(see for instance Example 3.2 4.(a)).To conclude this section, we analyze the relationship between local and global amoebas.Recall that, by Proposition 3.4, every global amoeba G has δ ( G ) ∈ { , } . On the otherhand, a local amoeba can have minimum degree arbitrarily large (see Example 3.2 item 3(a)). Interestingly, a local amoeba with minimum degree 0 or 1 is a global amoeba too,and the converse is true only when δ ( G ) = 0. We will prove the latter facts in the nextcorollary. Before this, we shall note that, in item 4 of Example 3.2, connected as well asnon-connected global amoebas with minimum degree one are presented which, in fact, arenot local amoebas. Corollary 3.9.
Let G be a graph minimum degree δ .(i) If δ ∈ { , } and G is a local amoeba, then G is a global amoeba.(ii) If δ = 0 , then G is a local amoeba if and only if G is a global amoeba.Proof. Let G be a local amoeba with δ ( G ) ∈ { , } . By Lemma 3.6, G ∪ K is a localamoeba and thus Theorem 3.7 yields that G is a global amoeba. This proves item (i) andthe “only if” part of item (ii). To prove the “if” part of item (ii), suppose that G is a globalamoeba with δ ( G ) = 0. Let X be the set of isolated vertices of G with | X | = q ≥ G (cid:48) = G − X . Hence, G = G (cid:48) ∪ qK . Since G is a global amoeba, Theorem 3.7 implies that G ∪ K is a local amoeba. But G ∪ K = G (cid:48) ∪ qK ∪ K = G (cid:48) ∪ ( q − K ∪ K and wecan use again Theorem 3.7 (with t=2) to conclude that G (cid:48) ∪ ( q − K ∪ K = G is a localamoeba, too. 13 .2 Proofs for the statements in Example 3.2 Now we can prove all statements of Example 3.2. We will use two ways of generating thesymmetric group S n : the one described in Observation 3.5 and the fact that (cid:104) σ, τ (cid:105) ∼ = S n ,where σ is an n -cycle and τ any transposition of two consecutive elements of the cycle.1.(a) The star K ,k − on k vertices is neither a local amoeba nor a global amoeba for k ≥ r -regular graph is not a local amoeba by Proposition 3.3 (ii) becausethe regularity implies it has only trivial feasible edge-replacements. An r -regulargraph is not a global amoeba, for r ≥ P k on k vertices is both a local and a global amoeba for k ≥ P k be defined on the vertex set { v , v , . . . , v k } with L ( P k ) = { , , . . . , ( k − k } . Consider the feasible edge-replacements ( k − k → k and2 3 → . . . k ) and (1 2) that generate S k . Thus, P k is a local amoeba and, by item (i) of Corollary 3.9, it is also a global amoeba.2.(b) The graph C ( k,
1) obtained from a cycle on k vertices by attaching a pendant vertex,for k ≥
3. Let C ( k,
1) be defined on the vertex set { v , v , . . . , v k +1 } with edges { v i v i +1 | i ∈ Z k } ∪ { v v k +1 } . Then 1 ( k + 1) → k ( k + 1) and ( k − k → ( k −
1) ( k + 1)are feasible edge-replacements that give the permutations (1 2 3 . . . k ) and ( k k + 1)that generate S k +1 . Thus, C ( k,
1) is a local amoeba and, item (i) of Corollary 3.9, itis also a global amoeba.3.(a) Let k ≥
4. The graph K − k obtained from the complete graph K k by deleting one edgeis a local amoeba by Proposition 3.3 (i) since K − k = K ∪ ( k − K is clearly a localamoeba. On the other hand, K − k is not a global amoeba because of Proposition 3.4.3.(b) Let C +5 be defined on the vertex set { v , v , v , v , v } with edges v v , v v , v v , v v , v v . Consider the feasible edge-replacements 4 5 → → S G +5 (5). Hence, Stab S G +5 (5) actstransitively on the set [4] and, since we also have (1 5) ∈ S G +5 by the feasible edge-replacement 1 2 → C +5 is a local amoeba by Observation 3.5.4.(a) For t ≥ k ≥ tP k is not a local amoeba by Proposition 3.3 because it hasonly trivial edge-replacements. However, the graph tP k is a global amoeba because ofTheorem 4.1 and the fact that P k is a local amoeba (see item 2(a) of this example).4.(b) Let T be the set of trivial feasible edge replacements of G . Then the set of feasibleedge-replacements of G is R G = T ∪{ → , → , → , → , → , → , → , → } and the set of permutations associated to thosefeasible edge-replacements is E G = { id, (2 3) , (2 4) , (1 2) , (5 9)(6 8) , (1 2)(5 9)(6 8) , (8 10) , (6 10) , (6 9)(7 8) } . E G , it is not difficult to see that (cid:104)E G (cid:105) = S G has two orbits, namely { , , , } and { , , , , , } (this can be also be deduced from seeing the picture of G in Figure 4). Since S G (cid:54) = S , G is not a local amoeba. However, since everyorbit in S G has a vertex of degree one, we deduce from Theorem 3.7(ii) that G isa global amoeba. In this section, we will give some constructions of amoebas that arise from smaller amoebas.In particular, we will be able to construct large and also dense amoebas, as well as amoebaswith large cliques and amoeba-trees with arbitrarily large maximum degree.The next theorem allows us to enlarge a global amoeba by means of taking a copy of aportion of its components where either an edge is added or deleted.
Theorem 4.1.
Let G = H (cid:48) ∪ H (cid:48)(cid:48) be a global amoeba, where H (cid:48) and H (cid:48)(cid:48) are vertex-disjointsubgraphs of G (where H (cid:48)(cid:48) can be possibly empty, meaning that G = H (cid:48) ) and such that E ( G ) = E ( H (cid:48) ) ∪ E ( H (cid:48)(cid:48) ) . Let H be a copy of H (cid:48) which is vertex disjoint from G . Then wehave the following facts.(i) If E ( H ) (cid:54) = ∅ , then, for any e ∈ E ( H ) , G ∪ ( H + e ) is a global amoeba.(ii) If E ( H ) (cid:54) = ∅ , then, for any e ∈ E ( H ) , G ∪ ( H − e ) is a global amoeba.Proof. We will give only the proof of item (i) as the one of (ii) can be deduced similarly.Let V ( G ) = { v , v , . . . , v n } , and V ( H (cid:48) ) = { v , v , . . . , v m } , where m ≤ n . Let V ( H ) = { v n +1 , v n +2 , . . . , v n + m } and e = v n + j v n + k ∈ E ( H ) for some j, k ∈ [ m ]. Assume, withoutloss of generality, that v n + i is the copy of v i in H , 1 ≤ i ≤ m . Then ( n + j ) ( n + k ) → j k is a feasible edge-replacement in G ∪ ( H + e ) and the permutation σ ∈ S n + m defined by σ ( i ) = n + i and σ ( n + i ) = i , for 1 ≤ i ≤ m , and σ ( i ) = i for m + 1 ≤ i ≤ n , is contained in S G ∪ ( H + e ) (( n + j ) ( n + k ) → j k ). Since S G ≤ S G ∪ ( H + e ) , we also have S G i ⊆ S G ∪ ( H + e ) i forany i ∈ [ n ]. If, in particular, i ∈ [ m ], then n + i ∈ S G ∪ ( H + e ) i as σ ( n + i ) = i . Since G is aglobal amoeba, we know by the equivalence of items (i) and (ii) of Theorem 3.7, that S G i ,and thus S G ∪ ( H + e ) i , contains an element l ∈ [ n ] such that deg G ∪ ( H + e ) ( v l ) = deg G ( v l ) = 1,hence G ∪ ( H + e ) is a global amoeba and we are done.The converse statements of this theorem are not valid. For item (i), we can take ( C k ∪ K ) ∪ C ( k,
1) that is a global amoeba by Theorem 4.1 (ii) because C ( k,
1) is a global amoeba(Example 3.2 2(b)). However, C k ∪ K is not a global amoeba because it has only trivialfeasible edge-replacements and it is nor complete nor empty (Proposition 3.3 (ii)). On theother hand, for item (ii), consider the graph C k ∪ P k , for k ≥ P k is a global amoeba, while C k is not.From Theorem 4.1 we can deduce the following. Corollary 4.2.
Let G be any connected graph. Then there is a global amoeba H having G as one of its components. roof. We will construct a global amoeba H by means of the following recursion. Let H = K . For i ≥
1, we do the following. If H i − (cid:54)∼ = G , then either there is one edge e ∈ E ( H i − ) such that the graph H i − + e is contained in G as a subgraph, or there is oneedge e ∈ E ( H i − ∪ K ) \ E ( H i − ) such that ( H i − ∪ K ) + e is contained in G as a subgraph.In the first case we set H i to be a copy of H i − + e , in the second case to be a copy of( H i − ∪ K ) + e . Since we add in each step a new edge and the obtained graph is alwayscontained in G as a subgraph, after m = e ( G ) steps, we will obtain a component H m ∼ = G .By means of m consecutive applications of Theorem 4.1 (i) (where sometimes H i − andsometimes H i − ∪ H plays the role of of H (cid:48) ) and since H = K is a global amoeba, itfollows that H = (cid:83) mi =0 H i is a global amoeba having one of its components isomorphic to G . As a consequence of Theorem 4.1, we obtain that there are global amoebas havingarbitrarily large chromatic and clique number. Corollary 4.3.
For any integer n , there is a global amoeba H with χ ( H ) = ω ( H ) = n . As we know, paths, the simplest trees one can imagine having only 1 and 2-degree ver-tices, are global amoebas. In this section, we will construct an infinite family of treesvia a Fibonacci-recursion which are global amoebas and which will have arbitrarily largemaximum degree (and by Proposition 3.4 vertices of all other possible degrees).
Lemma 4.4.
Let G be a graph on vertex set V = { v i | i ∈ [ n ] } . Let G = G (cid:48) ∪ G (cid:48)(cid:48) fortwo subgraphs G (cid:48) and G (cid:48)(cid:48) with respective vertex sets V (cid:48) and V (cid:48)(cid:48) . Let J (cid:48) and J (cid:48)(cid:48) be the setsof indexes of the vertices in V (cid:48) and V (cid:48)(cid:48) , respectively, and let I = J (cid:48) ∩ J (cid:48)(cid:48) . If there is a σ ∈ E G (cid:48) ∩ (cid:84) j ∈ I Stab S G (cid:48) ( j ) , then the permutation (cid:98) σ ( i ) = (cid:26) σ ( i ) , for i ∈ J (cid:48) \ Ii, for i ∈ J (cid:48)(cid:48) is in E G .Proof. Let σ ∈ E G (cid:48) ∩ (cid:84) j ∈ I Stab S G (cid:48) ( j ). Then there is a feasible-edge replacement rs → kl ∈ R G (cid:48) with r, s, k, l ∈ J (cid:48) . This edge-replacement gives a copy G (cid:48) σ of G (cid:48) that leaves the vertices v i with i ∈ I untouched, i.e. σ ( i ) = i for all i ∈ I . Then G = G (cid:48) ∪ G (cid:48)(cid:48) ∼ = G (cid:48) σ ∪ G (cid:48)(cid:48) = G (cid:98) σ .Hence, rs → kl is also a feasible edge-replacement in G and (cid:98) σ ∈ S G ( rs → kl ) ⊆ E G . Example 4.5.
The graph G depicted below in Figure 5 is built by the union of the graph G (cid:48) with index set J (cid:48) = { , , , , , } and the graph G (cid:48)(cid:48) with index set J (cid:48)(cid:48) = { , , , , , } .The edge-replacement → is feasible in G (cid:48) and we have that σ = (2 3) ∈ S G (cid:48) (12 → .Since σ = (2 3) ∈ E G (cid:48) ∩ (cid:84) i =4 , , Stab S G (cid:48) ( i ) , it follows by previous lemma that (cid:98) σ = (2 3) ∈ E G . Sketch for Example 4.5.
Let G be a graph on vertex set V = { v i | i ∈ [ n ] } and H another graph provided with aspecial vertex called the root of H . Let I = { i , i , . . . , i k } ⊆ [ n ]. We define G ∗ I H as thegraph obtained by taking G and k different copies H , H , . . . , H k of H and identifying theroot of H j with vertex v i j of G , for 1 ≤ j ≤ k (see Figure 6). Lemma 4.6.
Let G be a graph on vertex set V = { v i | i ∈ [ n ] } and H another graphof order m provided with a root. Let I = { i , i , . . . , i k } ⊆ [ n ] , N = n + k ( m − and let [ N ] = [ n ] ∪ (cid:83) k(cid:96) =1 J i (cid:96) be a partition of [ N ] such that | J i (cid:96) | = m − for all ≤ (cid:96) ≤ k . Let G ∗ I H consist of G and copies H i , H i , . . . , H i k of H such that V ( H i (cid:96) ) = { v i | i ∈ J i (cid:96) ∪ { i (cid:96) }} , for ≤ (cid:96) ≤ k . For any (cid:96), (cid:96) (cid:48) ∈ [ k ] , (cid:96) (cid:54) = (cid:96) (cid:48) , let ϕ i (cid:96) ,i (cid:96) (cid:48) : J i (cid:96) → J i (cid:96) (cid:48) be the bijection between givenby an isomorphism between H i (cid:96) and H i (cid:96) (cid:48) that sends v i (cid:96) to v i (cid:96) (cid:48) . If σ ∈ E G ∩ Stab S G ( I ) , thenthe permutation (cid:101) σ ( i ) = (cid:26) σ ( i ) , for i ∈ [ n ] ϕ i (cid:96) ,σ ( i (cid:96) ) ( i ) , for i ∈ J i (cid:96) , (cid:96) ∈ [ k ] is in E G ∗ I H .Proof. Let σ ∈ E G ∩ Stab S G ( I ). Then there is a feasible-edge replacement rs → kl ∈ R G with r, s, k, l ∈ [ n ]. This edge-replacement gives a copy G σ of G such that σ ( i ) ∈ I for all i ∈ I . Then ( G ∗ I H ) (cid:101) σ = G σ ∗ I H ∼ = G ∗ I H, implying that rs → kl is also a feasible edge-replacement in G ∗ I H and thus (cid:101) σ ∈ E G ∗ I H . Example 4.7.
Let G = v v v v v ∼ = P and H ∼ = K , + e , i.e. a star on three peakstogether with an edge joining two of the vertices of degree , where we designate one of thevertices of degree as the root of H . Let I = { , } , J = { , , , } , and J = { , , , } .Then → ∈ R G with σ = (1 4)(2 3) ∈ S G (4 5 → ∈ E G ∩ Stab S G ( { , } ) , and ϕ , = ϕ , = (2 3)(6 9)(7 10)(8 11) . It follows by Lemma 4.6 that (cid:101) σ = (1 4)(2 3)(6 9)(7 10)(8 11) ∈E G ∗ I H . Sketch of a graph G ∗ I H and of Example 4.7. We will describe a family of trees that are constructed via a Fibonacci recursion. Wedefine T = T = K . For i ≥
2, we define T i +1 as the tree consisting of one copy T of T i − and one copy T (cid:48) of T i , where a vertex of maximum degree of T is joined to a vertex ofmaximum degree of T (cid:48) by means of a new edge, see Figure 7. Observe that ∆( T i ) = i − i ≥
2, while n ( T i ) = 2 F i , being F i the i -th Fibonacci number. Note also that, for i ≥ T i has only one vertex of maximum degree, which we will call the root of T i . For the casethat i ≤
3, we will designate one of the vertices of maximum degree as the root of T i andthis will be the vertex that will be used to attach the new edge in the construction of T i +1 .Figure 7: Fibonacci amoeba-trees T i , 1 ≤ i ≤ Theorem 4.8. T i is a global amoeba for all i ≥ .Proof. Let T be a tree isomorphic to T i . Let J be the set of indexes of the vertices of T , i.e. V ( T ) = { v k | k ∈ J } and let c ∈ J such that v c has maximum degree in T . We will showby induction on i that there is a subset S ⊆ E T ∩ Stab S T ( c ) such that (cid:104) S (cid:105) acts transitivelyon J \ { c } .If i = 1 ,
2, there is nothing to prove. If i = 3, then T ∼ = P , say T = v v v v with c = 1. Then the feasible edge-replacements 34 →
24 and 13 →
14 give respectivelythe permutations (2 3) and (3 4), which act transitively on { , , } = J \ { c } . If i = 4,then let T be the tree built from the path v v v v ∼ = T and a T ∼ = K , given by v v ,18nd the edge v v joining both trees. Clearly, the only maximum degree vertex is v andthus c = 1. Then the feasible edge-replacements 34 →
24 and 13 →
14 give respectivelythe permutations (2 3) and (3 4), which together with the automorphism (3 5)(4 6), acttransitively on [5] \ { } = J \ { c } leaving c = 1 fixed.Now suppose that i ≥ i . Let T ∼ = T i +1 . Hence, | J | = 2 F i +1 . For a subset X ⊂ J , we define V X = { v x | x ∈ X } and T X = T [ V X ]. Let J = U ∪ W be a partition of J such that T U ∼ = T i − and T W ∼ = T i . Further, let U = A ∪ B and W = C ∪ D be partitions such that T A ∼ = T i − , T B ∼ = T i − , T C ∼ = T i − , and T D ∼ = T i − . By construction, v c is the root of T C .Let a, b, d ∈ J be such that v a , v b , v d are the roots of T A , T B , and T D , respectively. Noticethat v a v b v c v d is a path of length 4 in T . See Figure 8 for a sketch.Figure 8: Sketch of the tree T ∼ = T i +1 with its subtrees T U ∼ = T i − and T W ∼ = T i , and subsubtrees T A ∼ = T i − , T B ∼ = T i − , T C ∼ = T i − , and T D ∼ = T i − . By the induction hypothesis, there are subsets S U ⊆ E T U ∩ Stab S TU ( b ) and S W ⊆E T W ∩ Stab S TW ( c ) such that (cid:104) S U (cid:105) acts transitively on U \ { b } and (cid:104) S W (cid:105) acts transitivelyon W \ { c } . Let (cid:98) S U = { (cid:98) σ | σ ∈ S U } and (cid:98) S W = { (cid:98) σ | σ ∈ S W } with (cid:98) σ as in Lemma 4.4.Then, by precisely this lemma, (cid:98) S U , (cid:98) S W ⊆ E T . Moreover, the transitive action is inherited,i.e., (cid:104) (cid:98) S U (cid:105) acts transitively on U \ { b } and (cid:104) (cid:98) S W (cid:105) acts transitively on W \ { c } .Consider now the tree T ( B, D ) that is obtained by identifying all vertices from V B withvertex v b and all vertices from V D with vertex v d , i. e. we contract the sets V B and V D each to a single vertex (see Figure 9). Observe that ab → ad is a feasible edge-replacementin T ( B, D ) with τ = ( b d ) ∈ S T ( B,D ) ( ab → ad ), and that T ∼ = T ( B, D ) ∗ { b,d } T i − . Since T B ∼ = T D ∼ = T i − , there is a bijection ϕ : B → D given by an isomorphism between T B and T D such that ϕ ( b ) = d . Then, by Lemma 4.6, we have that ab → ad is a feasibleedge-replacement in T with (cid:101) τ ∈ S T ( ab → ad ) such that (cid:101) τ ( i ) = ϕ ( i ) , for i ∈ Bϕ − ( i ) , for i ∈ Di, else,and which fulfills that (cid:101) τ ∈ E T . Moreover, (cid:101) τ leaves c fixed and so (cid:101) τ ∈ E T ∩ Stab S T ( c ). Nowwe define S = (cid:98) S U ∪ (cid:98) S W ∪ { (cid:101) τ } . (cid:104) (cid:98) S U (cid:105) acts transitively on U \ { b } , and (cid:104) (cid:98) S W (cid:105) acts transitively on W \ { c } , these twosets together with (cid:101) τ generate a group (cid:104) S (cid:105) that acts transitively on J \ { c } .Hence, we have shown that if T ∼ = T i , for any i ≥