On the number of non-isomorphic matroids

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Advances in Applied Mathematics 33 (2004) 733–746

www.elsevier.com/locate/yaam

On the number of non-isomorphic matroids

Manoel Lemos1

Departamento de Matemática, Universidade Federal de Pernambuco, Cidade Universitária, Recife,Pernambuco, 50740-540, Brazil

Received 13 November 2003; accepted 4 July 2004

Available online 3 September 2004

Abstract

Let f (n) denote the number of non-isomorphic matroids on ann-element set. In 1969, Welsconjectured that, for all non-negative integersm andn, f (m + n) � f (m)f (n). In this paper, weprove this conjecture. 2004 Elsevier Inc. All rights reserved.

1. Introduction

The main result proved in this paper is the following theorem which was conjectured bWelsh [6].

Theorem 1. If m and n are non-negative integers, then f (m + n) � f (m)f (n).

This conjecture is listed as a problem in Oxley [5]. Blackburn, Crapo, and Higgfound all non-isomorphic simple matroids on at most eight elements and they said thconjecture is “surely correct” but it has still not been proved. The structure of the prquite simple. We construct an injective functionΨ :Fm ×Fn → Fm+n, whereFk denotesthe set of non-isomorphic matroids on ak-element set. To constructΨ , we define a newoperation between two matroids and, in Lemma 4, we characterize when we cannot r

E-mail address: [email protected] The author was partially supported by CNPq (Grant No. 302195/02-5) and ProNEx/CNPq (Grant N

664107/97-4).

0196-8858/$ – see front matter 2004 Elsevier Inc. All rights reserved.doi:10.1016/j.aam.2004.07.001

734 M. Lemos / Advances in Applied Mathematics 33 (2004) 733–746

thesed 7,

thesess theture isma 2,we use

sing

it or

medelled

d

y

m

these matroids from the result of this operation. This happens in very few cases. Infew cases, we define another operation between these matroids, and in Lemmas 6 anwe prove that we can recover the matroids from the result of this operation. To definetwo operations, we need to add an element freely in a matroid. In Lemma 3, we discustructure of the tree decomposition of a matroid that has a free element. The strucquite simple and it will be fundamental in the proof of the subsequent lemmas. In Lemwe characterize the matroids that has both a free and cofree element. In this paper,the notation and terminology set by Oxley in [5].

An important tool in the proof of the main theorem is the following idea of decompoa connected matroidM. Assume|E(M)| � 3. A tree decomposition of M is a treeT withedges labelled bye1, e2, . . . , ek−1 and vertices labelled by matroidsM1,M2, . . . ,Mk suchthat

(i) eachMi is a 3-connected matroid having at least four elements or is a circucocircuit with at least three elements;

(ii) E(M1) ∪ E(M2) ∪ · · · ∪ E(Mk) = E(M) ∪ {e1, e2, . . . , ek−1};(iii) if the edgeei joins the verticesMj1 andMj2, thenE(Mj1) ∩ E(Mj2) = {ei};(iv) if no edge joins the verticesMj1 andMj2, thenE(Mj1) ∩ E(Mji ) is empty;(v) M is the matroid that labels the single vertex of the treeT/e1, e2, . . . , ek−1 at the

conclusion of the following process: contract the edgese1, e2, . . . , ek−1 of T one byone in this order; whenei is contracted, its ends are identified and the vertex forby this identification is labelled by the 2-sum of the matroids that previously labthe ends ofei .

Cunningham and Edmonds [2] proved the following result.

Theorem 2. Every connected matroid M has a tree decomposition T (M) in which no twoadjacent vertices are both labelled by circuits or are both labelled by cocircuits. Further-more, the tree T (M) is unique to within relabelling of its edges.

We shall callT (M) the canonical tree decomposition of M. Let M be a connectematroid andT be a tree decomposition ofM. A connected subgraphH of T inducesa subsetX of E(M) if X is the union, over all verticesMj of H , of E(Mj) ∩ E(M).Each edgee of T determines a partition ofE(M) into the subsetsXe1 andXe2 that areinduced by the components ofT − e. We shall say that the edgee displays the partition{Xe1,Xe2} of E(M). Now let M ′ be a vertex ofT that is a circuit or a cocircuit. We sathatM ′ displays a partition{X,Y } of E(M) if every subset ofE(M) that is induced by acomponent ofT −M ′ lies entirely in eitherX or Y . We need the next result of Cunninghaand Edmonds [2]:

Lemma 1. Let M be a connected matroid and {X1,X2} be a partition of E(M) such thatmin{|X1|, |X2|} � 2. Then the following statements are equivalent:

(i) {X1,X2} is a 2-separation of M;

M. Lemos / Advances in Applied Mathematics 33 (2004) 733–746 735

or

t

(ii) T (M) has an edge or a vertex that displays {X1,X2} where, in the latter case, thevertex is labelled by a circuit or a cocircuit.

For a short proof of this result, see Lemos and Oxley [4].

2. Proof of the main result

It is possible that the next two lemmas are known but we do not know a reference fthem.

Lemma 2. Let e be an element of a connected matroid M such that |E(M)| � 2. If e is freein both M and M∗, then M is a uniform matroid.

Proof. As e is free inM, it follows that

Ce(M) = {e ∪ B: B ∈ B(M\e)} and Ce

(M∗) = {

e ∪ C∗: C∗ ∈ C(M∗/e

)},

whereCh(H) denotes the set of circuits of a matroidH that contain the elementh. But eis also free inM∗ and soCe(M) = {e ∪ C: C ∈ C(M/e)}. Thus

C(M/e) = B(M\e). (1)

Now, we show thatM\e is uniform. IfX ⊆ E(M\e) and|X| = r(M), thenX is dependenin M/e. Hence there is a circuitC of M/e such thatC ⊆ X. By (1), C is a basis ofM\eand soC = X. Hence every subset ofE(M\e) with r(M) elements is a basis ofM\e. ThusM\e is a uniform matroid. AsM is obtained fromM\e by addinge freely, it follows thatM is uniform. �

For a matroidM and an elemente not belonging toM, we denote byM + e the matroidobtained fromM by addinge freely. Denote by loop(M) the set of loops ofM.

Lemma 3. Suppose that M is a matroid such that r(M) � 1. If e is not an element of M ,then loop(M + e) = loop(M) and N = (M + e)\loop(M) is connected. Moreover, when n

is the number of connected components of M\loop(M) and K labels the vertex of T (N)

having e as an element, then,

(i) when r(M) = 1, then K is a cocircuit and K is the unique vertex of T (N);(ii) when r(M) � 2 and n = 1, then K is a 3-connected matroid having at least four

elements and, for each non-trivial parallel class P of M , there is an edge of T (N)

labeled by eP which is incident to vertices labeled by K and by a cocircuit withground set P ∪ eP . Moreover, these are the only edges of T (N); and

(iii) when r(M) � 2 and n � 2, then K is a circuit having n + 1 elements and, for everyconnected component H of M such that |E(H)| � 2, there is eH ∈ E(K) such thateH labels an edge of T (N) incident with the vertex labeled by K , and the connected


t

,llel

-

component of T (N) − eH which does not contain the vertex labeled by K is equalto T (H + eH ). Moreover, these are the only edges of T (N) incident with the vertexlabeled by K .

Proof. We divide the proof of this lemma in some steps.

Step 1. Let {X,Y } be a partition ofE(M). (By definition,X �= ∅ andY �= ∅.) Then

(1) {X ∪ e,Y } is a 1-separation ofM + e if and only if rM(Y ) = 0.(2) {X ∪ e,Y } is an exact 2-separation ofM + e if and only if |Y | � 2 and

(a) {X,Y } is a 1-separation ofM such thatrM(Y ) �= 0; or(b) Y − loop(M) is a non-empty subset of a parallel class ofM.

Observe that

rM+e(X ∪ e) + rM+e(Y ) − r(M + e) = [rM(X) + rM(Y ) − r(M)

] + δ, (2)

whereδ = 0, whenrM(X) = r(M), andδ = 1, whenrM(X) < r(M). Thus{X ∪ e,Y } is a1-separation ofM + e if and only if rM(X)+ rM(Y )− r(M) = δ = 0. But this is equivalento rM(X) = r(M) andrM(Y ) = 0. Moreover,{X ∪ e,Y } is an exact 2-separation ofM + e

if and only if |Y | � 2 and

(c) rM(X) + rM(Y ) − r(M) = 0 andδ = 1; or(d) rM(X) + rM(Y ) − r(M) = 1 andδ = 0.

Note that (c) is equivalent to (a) and (d) implies (b).

By Step 1, loop(M + e) = loop(M) andN is a connected matroid. Ifr(M) = 1, then (i)follows. We may assumer(M) � 2.

Step 2. If M\ loop(M) is connected, then (ii) follows.Let P1, . . . ,Pk be the non-trivial parallel classes ofM and let {e1, . . . , ek} be a k-

element set disjoint ofE(M + e). There are matroidsN0,N1, . . . ,Nk such thatE(N0) =[E(N) − (P1 ∪ · · · ∪ Pk)] ∪ {e1, . . . , ek}, for i ∈ {1, . . . , k}, E(Ni) = Pi ∪ ei , Ni is a cocir-cuit andN is the 2-sum ofN0,N1, . . . ,Nk . AsN\e = M\loop(M) is connected, it followsby Step 1, that each 2-separation ofN has one of its sets contained in a non-trivial paraclass ofM and soN0 is a 3-connected matroid having at least four elements. ThusK = N0and (ii) follows.

By Step 2, we may assume thatM\ loop(M) is not connected. LetM1, . . . ,Mn be theconnected components ofM\ loop(M). Suppose also that

∣∣E(M1)

∣∣ � · · · � ∣

∣E(Mm)∣∣ >

∣∣E(Mm+1)

∣∣ = · · · = ∣

∣E(Mn)∣∣ = 1,

for some integerm such that 0� m � n. Let {f1, . . . , fm} be anm-element set disjoint of E(M + e). For i ∈ {1, . . . ,m}, let M ′ be the matroid obtained fromN/[E(N) −
i


ithd

et

nd

is

in

(E(Mi)∪e)] by relabellinge asfi . Note thatM ′i = Mi +fi . LetM ′

0 be the(n+1)-elementcircuit with ground set{e, f1, . . . , fm} ∪ E(Mm+1) ∪ · · · ∪ E(Mn). By [3, Lemma 2.10],N is the 2-sum ofM ′

0,M′1, . . . ,M

′m. This 2-sum decomposition can be associated w

a starT with m + 1 vertices having center labeled byM ′0, the other vertices labele

by M ′1, . . . ,M

′m and, for i ∈ {1, . . . ,m}, the edge incident withM ′

i labeled byfi . Fori ∈ {1, . . . ,m}, let Ki be the matroid labeling a vertex ofT (M ′

i ) such thatfi ∈ E(Ki).As M ′

i\fi = Mi is connected, it follows thatKi is not a circuit. IfT ′ is the star obtainedfrom T by relabeling the terminal verticesM ′

1, . . . ,M′m respectively byK1, . . . ,Km, then

T (N) is the union ofT ′, T (M ′1), . . . , T (M ′

m). Thus (iii) follows withK = M ′0. �

Note that in (iii) of Lemma 3, the structure ofT (H + eH ), whereH is a connectedcomponent ofN having at least two elements, is described in (i), whenr(H) = 1, or in (ii),whenr(H) � 2. Thus this lemma describes completely the structure of the treeT (N).

Now, we define a new operation between matroids with disjoint ground sets. LM

andN be matroids such thatE(M) ∩ E(N) = ∅, |E(M) − loop(M)| � 2 and|E(N) −loop(N∗)| � 2. If e /∈ E(M) ∪ E(N), then we defineM N as

M N = (M + e) ⊕2(N∗ + e

)∗.

(Remember that(N∗ + e)∗ is the free coextension ofN .) Note that(M N)∗ = N∗ M∗.By Lemma 3,

loop(M N) = loop(M) and loop([M N]∗) = loop

(N∗). (3)

Moreover,

[M\ loop(M)

] [N\ loop

(N∗)] = [M N]\[loop(M) ∪ loop

(N∗)]. (4)

For a matroidH and positive integersr, s and n such that max{r, s} � n, we say that(M,N) is a(r, s, n,H)-pair providedM andN are matroids with disjoint ground sets aM\loop(M) ∼= Us,n andN\ loop(N∗) ∼= Ur−1,n ⊕ H .

Lemma 4. If M,N,M ′ and N ′ are matroids such that E(M) ∩ E(N) = E(M ′) ∩E(N ′) = ∅, |E(M)| = |E(M ′)|, |E(N)| = |E(N ′)| and M N ∼= M ′ N ′, then

(i) M ∼= M ′ and N∗ ∼= N ′ ∗; or(ii) there is a matroid H and positive integers r, s and n such that (M,N) is a (r, s, n,H)-

pair and (M ′,N ′) is a (s, r, n,H)-pair; or(iii) there is a matroid H and positive integers r, s and n such that (N∗,M∗) is a

(r, s, n,H)-pair and (N ′ ∗,M ′ ∗) is a (s, r, n,H)-pair.

(At some steps of the proof of this lemma, we use the expressiontaking the dual. Thismeans that we replace(M,N,M ′,N ′) by (N∗,M∗,N ′ ∗,M ′ ∗). Note that the hypothesof this lemma still holds after this modification since[M N]∗ = N∗ M∗. Moreover,the conclusion described in (i) is invariantunder duality and the conclusions described(ii) and (iii) form a dual pair.)


e

c-t

-el

Proof. Suppose this result is not true and choose a counter-example(M,N,M ′,N ′) sothat|E(M)| + |E(N)| is minimum. First, we prove that

loop(M) = loop(N∗) = loop

(M ′) = loop

(N ′ ∗) = ∅. (5)

By (3),

∣∣loop(M)

∣∣ = ∣

∣loop(M ′)∣∣ and

∣∣loop

(N∗)∣∣ = ∣

∣loop(N ′ ∗)∣∣. (6)

By (4),

[M\ loop(M)

] [N\ loop

(N∗)] ∼= [

M ′\ loop(M ′)] [

N ′\ loop(N ′ ∗)]. (7)

If | loop(M)| + | loop(N∗)| �= ∅, then, by the choice of(M,N,M ′,N ′), (6) and (7),the result holds for(M\ loop(M),N\ loop(N∗),M ′\ loop(M),N ′\ loop(N ′ ∗)); a contra-diction by (3) and (6). Thus (5) follows. By Lemma 3,M N is connected. SincM N ∼= M ′ N ′, we can relabel the elements ofM ′ andN ′ so thatM N = M ′ N ′.We may also assume thatT (M N) = T (M ′ N ′).

By definition, we obtainM N as the 2-sum ofM + e and [N∗ + e]∗, for someelemente not belonging toE(M)∪E(N). LetHe andKe be the matroids labelling respetively a vertex ofT (M + e) andT ([N∗ + e]∗) such thate ∈ E(He)∩E(Ke). (Suppose thathe labels of the edges ofT (M + e) andT ([N∗ + e]∗) are different.) HenceT (M N) isobtained by

(T1) making the union ofT (M + e) andT ([N∗ + e]∗);(T2) adding an edge labelled bye joining He to Ke ; and(T3) whenHe andKe are both circuits or both cocircuits, contractinge and labelling the

new vertex byHe ⊕2 Ke.

By this construction, when the 2-separation{E(M),E(N)} of M N is displayed by andedge ofT (M N), then this edge has label equal toe. Now, we prove thate does notdisplay also the 2-separation{E(M ′),E(N ′)} of M N . If e displays{E(M ′),E(N ′)},then

(a) M + e = M ′ + e and[N∗ + e]∗ = [N ′ ∗ + e]∗; or(b) M + e = [N ′ ∗ + e]∗ and[N∗ + e]∗ = M ′ + e.

We have a contradiction, when (a) happens, sinceM = M ′ andN = N ′. Thus (b) holds.As e is both free and cofree inM + e, it follows, by Lemma 2, thatM ∼= Us,n andN ′ ∼= Us−1,n, for 1 � s � n. Similarly, M ′ ∼= Ur,n andN ∼= Ur−1,n, for 1 � r � n. Thus(ii) follows for the empty matroidH ; a contradiction and soe does not display the 2separation{E(M ′),E(N ′)}. If the 2-separation{E(M ′),E(N ′)} is displayed by an edgof T (M N), then we denote this edge bye′. Observe thate or e′ or both may not labeedges ofT (M N). But when bothe ande′ label edges ofT (M N), we have that

e �= e′. (8)


ynt

, iff

e

3,

(9)

e may

t

Now, we prove an auxiliary lemma.

Lemma 5. If Ke is a cocircuit and e′ ∈ E(Ke), then there is a matroid H and positiveintegers r and n such that r � n, M ′ ∼= Ur,n and N ∼= Ur−1,n ⊕ H .

By (8), e′ labels an edge ofT ([N∗ + e]∗) incident with the vertex labeled bKe. As |E(M)| = |E(M ′)|, it follows that T (M ′ + e′) is the connected componeof T ([N∗ + e]∗) − e′ that does not include the vertex labeled byKe. By the dualof Lemma 3(iii), there is a connected componentK of N such thatT (M ′ + e′) =T ([K∗ + e′]∗) and soM ′ + e′ = [K∗ + e′]∗. Thuse′ is both free and cofree inM ′ + e′.Hence, by Lemma 2,M ′ + e is a uniform matroid and so there are positive integersr andn such thatr � n andM ′ ∼= Ur,n. Observe thatK ∼= Ur−1,n. ThusN ∼= Ur−1,n ⊕ H , forsome matroidH , and so Lemma 5 follows.

Next, we show that:

If Ke is a cocircuit, thene′ /∈ E(Ke). (9)

Assume thate′ ∈ E(Ke). By Lemma 5, there is a matroidH and positive integersr andn

such thatr � n, M ′ ∼= Ur,n andN ∼= Ur−1 ⊕ H . We have two cases to consider. Firste is an edge ofT (M N), thenKe = K ′

e′ , whereK ′e′ is the matroid labelling a vertex o

T ([N ′ ∗ + e′]∗) havinge′ as an element. Ase ∈ E(K ′e′), it follows, by Lemma 5, that ther

is a matroidH ′ and a positive integers such thats � n,M ∼= Us,n andN ′ ∼= Us−1 ⊕ H ′.Note thatH = H ′ because these matroids are equal toK/{e, e′}, whereK is the matroidsuch thatT (K) is equal to the connected component ofT (M N) − {e, e′} having thevertex labeled byKe. We have a contradiction because(ii) follows. Now, consider thesecond case. Ife is not an edge ofT (M N), thenHe is a cocircuit and so, by Lemmar(M) = 1. ThusM ∼= U1,n. Note thatT ([N ′ ∗ + e′]∗) is obtained fromT ([N∗ + e]∗) by

(T4) relabelling the vertex incident withe′ other thanKe by a cocircuitC∗ with groundsetE(M) ∪ e;

(T5) relabelling the edgee′ by e; and(T6) contracting the edgee and labelling the vertex created byKe ⊕2 C∗.

ThusN ′ ∼= U0,n ⊕ H . Again, we arrive at a contradiction because (ii) follows. Thusholds.

Now, we show that, in the construction ofT (M N) from T (M +e) andT ([N∗ +e]∗),(T3) does not happen. Assume the contrary. Taking the dual, when necessary, wsuppose that bothHe andKe are cocircuits. By Lemma 3,r(M) = 1 and soHe = M + e.First, we prove that

the 2-separation{E

(M ′),E

(N ′)} is not displayed by the vertexHe ⊕2 Ke. (10)

If He ⊕2 Ke displays{E(M ′),E(N ′)}, thenr(M ′) = 1. Since|E(M ′)| = |E(M)|, it fol-lows, by (5), thatM ′ ∼= M. Observe thatN ′ ∼= N becauseT ([N∗ +e]∗) andT ([N ′ ∗ +e′]∗)are obtained fromT (M N) by ralabellingHe ⊕2 Ke by a cocircuit having ground se


ir-y

.ef

)

its

fls ae,

,

)

l, thereed

e

es-

al of

equal respectively to[E(He ⊕2 Ke) − E(M)] ∪ e and [E(He ⊕2 Ke) − E(M ′)] ∪ e′;a contradiction and so (10) follows. By (10),He ⊕2 Ke labels a vertex ofT (M ′ + e′)or T ([N ′ ∗ + e′]∗). As |E(He ⊕2 Ke)| > |E(M)| + 1, it follows thatHe ⊕2 Ke labels avertex ofT ([N ′ ∗ + e′]∗). By the dual of Lemma 3,e′ must be an element of the coccuit He ⊕2 Ke. Hencee′ labels an edge ofT (M N) adjacent to the vertex labeled bHe ⊕2 Ke . As He = M + e, it follows, by (8), thate′ ∈ E(Ke); a contradiction to (9)Hence (T3) does not occur. Thus the 2-separation{E(M),E(N)} is displayed by an edgof T (M N). Similarly, the 2-separation{E(M ′),E(N ′)} is displayed by and edge oT (M N). Hencee ande′ label edges ofT (M N).

Next, we show that

M or N is connected. (11

Suppose (11) does not hold. AsM andN are not connected, it follows, by Lemma 3 anddual, thatHe is a circuit andKe is a cocircuit. Observe that bothHe andKe label verticesof T (M ′ + e′) or of T ([N ′ ∗ + e′]∗), since{E(M ′),E(N ′)} is displayed by an edge oT (M N) different of e. By Lemma 3 and its dual, a circuit or a cocircuit that labevertex ofT (M ′ + e′) or of T ([N ′ ∗ + e′]∗) containse′ or labels a terminal vertex. Hencby (8),He or Ke does not containe′ and soHe or Ke is a terminal vertex ofT (M ′ + e′) orT ([N ′ ∗ + e′]∗). Taking the dual, when necessary, we may assume thatHe does not containe′ and soHe is a terminal vertex ofT (M ′ + e′) or T ([N ′ ∗ + e′]∗). HenceHe is a terminalvertex ofT (M N). ThusM + e = He and sor∗(M) = 0. By (8), e does not displaythe 2-separation{E(M ′),E(N ′)} and soHe is not the unique vertex ofT (M ′ + e′) or ofT ([N ′∗ +e′]∗). ThusHe is adjacent to a vertex labeled by the cocircuitKe in T (M ′ +e′) orT ([N ′∗ + e′]∗) becauseHe is adjacent only toKe in T (M N). By the dual of Lemma 3e′ ∈ E(Ke); a contradiction to (9) and so (11) follows.

Now, we prove that

M or N is not connected. (12

Suppose that bothM andN are connected. If min{r(M), r(N∗)} � 2, thenKe andHe

are 3-connected matroids having at least four elements. By Lemma 3 and its duais no edge ofT (M ′ + e′) or T ([N ′ ∗ + e′]∗) joining two vertices labelled by 3-connectmatroids with at least four elements. Thuse = e′; a contradiction to (8). Hencer(M) = 1or r(N∗) = 1. Taking the dual, when necessary, we may assume thatr(M) = 1 and soHe = M + e is a cocircuit. Ase does not display{E(M ′),E(N ′)}, by (8), it follows thatHe labels a vertex ofT ([N ′ ∗ + e′]∗) because|E(He)| = |E(M)| + 1 = |E(M ′)| + 1. Bythe dual of Lemma 3,e′ ∈ E(He) becauseHe is a cocircuit; a contradiction to (8), sincHe is adjacent only toe in T (M N).

From (11) and (12), exactly one ofM andN is connected. Taking the dual, when necsary, we may assume thatM is connected andN is not connected. HenceKe is a cocircuit.As the 2-separation{E(M),E(N)} is displayed by an edge ofT (M N), it follows, byLemma 3, thatHe is a 3-connected matroid having at least four elements. By the duLemma 3,e′ is adjacent toKe or Ke is a terminal vertex ofT (M N). By (9), Ke is aterminal vertex ofT (M N). HenceT (M N) is a star havingHe as its center. Ase �= e′,


an

,,

con-theer

by (8), it follows thate ande′ are incident toHe and so{e, e′} ⊆ E(He). Observe thatKe′labels a terminal vertex ofT (M N) because|E(N)| = |E(N ′)|, |E(N)| = |E(Ke)| + 1andKe label a vertex of the connected component ofT (M N)−e′ that containsHe. ThusKe andKe′ are both cocircuits with cardinality|E(N)| + 1. In particular,N ∼= N ′. Let H ′′be a matroid such thatT (H ′′) is equal to the connected component ofT (M N) − {e, e′}havingHe as one of its vertices. Note thatM + e = H ′′ ⊕2 Ke′ andM ′ + e′ = H ′′ ⊕2 Ke.Thus bothe ande′ are free elements ofH ′′. Hence there is an automorphism ofH ′′ thatfixesE(H ′′) − {e, e′} and permutese with e′. This automorphism can be extended toisomorphism betweenM + e andM ′ + e′, sinceM + e andM ′ + e′ are obtained fromH ′′by replacing respectivelye′ ande by a parallel class having as elementsE(Ke) − e andE(K ′

e) − e′. ThusM ∼= M ′. We arrive to the final contradiction and the result follows.�We define a new operationM �N , for a (r, s, n,H)-pair (M,N), where H is a

matroid andr, s and n are positive integers such that max{r, s} � n, as follows: if H

is the empty matroid, thenM �N = M ⊕ N ; and if H is not the empty matroidthen M �N = M ⊕ [N∗ + e]∗\e. Observe that, whenH is not the empty matroidloop(M �N) = loop(M), loop([M �N]∗) ⊇ loop(N∗) and

[M\ loop(M)

]�

[N\ loop

(N∗)] = [

M �N ′]∖[loop(M) ∪ loop

(N∗)]. (13)

Lemma 6. Let H and H ′ be non-empty matroids and let r , r ′, s, s′, n and n′ be pos-itive integers such that max{r, s} � n,max{r ′, s′} � n′ and 2 � min{n,n′}. If (M,N) isa (r, s, n,H)-pair and (M ′,N ′) is a (r ′, s′, n′,H ′)-pair such that |E(M)| = |E(M ′)|,|E(N)| = |E(N ′)| and M �N ∼= M ′ �N ′, then M ∼= M ′ and N ∼= N ′.

Proof. Suppose this result is not true and choose a counter-example(M,N,M ′,N ′) sothat|E(M)|+ |E(N)| is minimum. By the dual of Lemma 3, the matroidKe havinge as anelement and labelling a vertex ofT ([N∗ + e]∗\ loop(N∗)) is a cocircuit and so[N∗ + e]∗\[e ∪ loop(N∗)] is connected. AsE([N∗ + e]∗\[e ∪ loop(N∗)]) = E(N) − loop(N∗), itfollows that [N∗ + e]∗\[e ∪ loop(N∗)] is the connected component ofM �N with themaximum number of elements. Hence

[N∗ + e

]∗∖[e ∪ loop

(N∗)] ∼= [

N ′ ∗ + e]∗∖[

e ∪ loop(N ′ ∗)]. (14)

By the choice of (M,N,M ′,N ′) and (13), loop(M) = loop(M ′) = loop(N∗) =loop(N ′ ∗) = ∅. ThusM ∼= M ′. In particular,

n = n′. (15)

SinceM �N ∼= M ′ �N ′, we can relabel the elements ofM ′ andN ′ so thatM �N =M ′ �N ′. HenceT ([N∗ +e]∗\[e∪ loop(N∗)]) = T ([N ′ ∗ +e]∗\[e∪ loop(N ′ ∗)]). We arriveat a contradiction by proving that it is possible to recover up to isomorphism thenected components ofN from T ([N∗ + e]∗\e). As these connected components areconnected components of[N∗ + e]∗/e, it is enough to prove that it is possible to recovT ([N∗+e]∗) fromT ([N∗+e]∗\e) up to isomorphism. But we obtainT ([N∗+e]∗\e) fromT ([N∗ + e]∗),


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oe new

esn twor

(a) when|E(Ke)| � 4, by relabellingKe by Ke\e; or(b) when |E(Ke)| = 3, sayE(Ke) = {e, a, b} (for x ∈ E(Ke), Hx denotes the matroi

different ofKe that labels a vertex ofT ([N∗ + e]∗) and containsx, when it exists),(b1) by contractingb and relabellingb by a in Hb, when bothHa andHb exist and

Ha or Hb is not a circuit; or(b2) by contractinga andb and relabelling the vertex obtained after the identificati

of the end vertices ofa andb by a circuit having[E(Ha) − a] ∪ [E(Hb) − b] asground set, when bothHa andHb are circuits; or

(b3) whenHa or Hb does not exist, sayHa , by contractingb and relabellingb bya in Hb. (Note that, in this case,b labels an edge ofT ([N∗ + e]∗), otherwise|E(N)| = 2.)

Observe that (a) is reversible becauseKe\e is a cocircuit that labels a vertex oT ([N∗ + e]∗\e) and so, by the dual of Lemma 3, it is the unique cocircuit labeling atex of T ([N∗ + e]∗\e). HenceT ([N∗ + e]∗) is obtained fromT ([N∗ + e]∗\e) by addinge to the ground set of the unique vertex labeled by a cocircuit. Thus (b) occurs. Ihappens, then|E(H)| = 1 andr > 1. (Remember thatn � 2, by hypothesis.) Moreove[N∗ + e]∗\e ∼= Ur,n+1 and soN ∼= Ur−1,n ⊕U0,1. ThusN ∼= N ′ because, by (14) and (15N ′ is obtained from a matroid isomorphic toUr,n+1 by adding an elemente′ in parallel toone of its elements and by doing the contraction ofe′. Hence (b1) or (b2) occurs. If (b2holds, then[N∗ + e]∗\e is an[n + |E(H)|]-element circuit because every vertex labeby a circuit inT ([N∗ + e]∗) is terminal. ThusN ∼= Un−1,n ⊕ U|E(H)|−1,|E(H)|. By (14)and (15),N ′ ∼= N . Thus (b1) holds. We have two cases to consider. First, ifHa andHb

are both 3-connected matroids with at least four elements, then by the dual of Lema is the unique edge ofT ([N∗ + e]∗\e) that joins two vertices labeled by 3-connecmatroids with at least four elements, since this does not happen inT ([N∗ + e]∗). In thiscase,T ([N∗ +e]∗) is obtained fromT ([N∗ +e]∗\e) by subdividing the edge that joins twvertices labeled by 3-connected matroids with at least four elements and labelling thvertex by a triad that containse. Thus we may assume thatHa or Hb is a circuit, sayHb.HenceT ([N∗ + e]∗\e) is a star havingHa labeling its center and all the other verticlabelled by circuits, by the dual of Lemma 3. Moreover, when this star has more thavertices, thenE(Hb) = E(N) − [E(H) ∪ loop(N∗)] and so|E(Hb)| = n. We also recoveT ([N∗ + e]∗) from it. To do this, it is enough to subdivide an edgec such thatc is cofreein Ha and|E(Hc)| = |E(Hb)| and to label the new vertex by a triad that containse. Withthis we arrive at the final contradiction and the result follows.�

Lemma 7. Let H and H ′ be non-empty matroids and let r, r ′, s, s′, n and n′ be pos-itive integers such that max{r, s} � n, max{r ′, s′} � n′ and 2 � min{n,n′}. If (M,N)

is a (r, s, n,H)-pair and (N ′ ∗,M ′ ∗) is a (r ′, s′, n′,H ′)-pair such that |E(M)| =|E(M ′)|, |E(N)| = |E(N ′)| and M �N ∼= [N ′ ∗ �M ′ ∗]∗, then s = n and s′ = n′. More-over, M �N ∼= [N ′ ∗ �M ′ ∗]∗ has just one connected component with at least two elements.


r-

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e

sk

Proof. Suppose this result is not true and choose a counter-example(M,N,M ′,N ′) sothat|E(M)| + |E(N)| is minimum. Hences < n or s′ < n′. By definition,

[N ′ ∗ �M ′ ∗]∗ = [

N ′ ∗ ⊕ ((M ′ ∗)∗ + e

)∗\e]∗ = N ′ ⊕ [(M ′ + e

)/e

].

By the dual of Lemma 3, the matroidKe having e as an element and labelling a vetex of T ([N∗ + e]∗\loop(N∗)) is a cocircuit and so[N∗ + e]∗\[e ∪ loop(N∗)] is con-nected. AsE([N∗ + e]∗\[e ∪ loop(N∗)]) = E(N) − loop(N∗), it follows that[N∗ + e]∗\[e ∪ loop(N∗)] is the connected component ofM �N with the maximum number of elements. Similarly,[M ′ + e]/[e ∪ loop(M ′)] is the connected component of[N ′ ∗ �M ′ ∗]∗with the maximum number of elements. Hence

[N∗ + e

]∗∖[e ∪ loop

(N∗)] ∼= [

M ′ + e]/[

e ∪ loop(M ′)]. (16)

Moreover, these matroids haven + |E(H)| = n′ + |E(H ′)| elements. Ass < n or s′ < n′,it follows thatM\ loop(M) or N ′\ loop(N ′ ∗) is a connected matroid having respectiveln

or n′ elements. As min{n,n′} � 2 andM �N ∼= [N ′ ∗ �M ′ ∗]∗ has at most two connectecomponents with more than one element, one of them being described in (16), it fothatM\ loop(M) ∼= N ′\loop(N ′ ∗) is connected. In particular,n = n′. By (16), |E(H)| =|E(H ′)|. Thus

loop(M �N) = loop(M) and loop([

N ′ ∗ �M ′ ∗]∗) = loop(M ′)

and so| loop(M)| = | loop(M ′)|. As | loop(M)| = |E(M)|−n and| loop(M ′)| = |E(M ′)|−(n′ + |E(H ′)|), it follows that

∣∣E(M)

∣∣ − n = ∣

∣E(M ′)∣∣ − (

n′ + ∣∣E

(H ′)∣∣).

We have a contradiction because|E(M)| = |E(M ′)| and n = n′. The first part of thislemma follows. The second part of it is straightforward.�

Observe that Theorem 1 is a consequence of the next result, when we apply it to thclass of all matroids.

Theorem 3. Let F be a class of matroids closed under duality, isomorphisms, minors,1-sums, 2-sums and free extensions. If m and n are non-negative integers, then

f (m + n) � f (m)f (n),

where f (k) denotes the number of non-isomorphic matroids belonging to F whose groundset has k elements.

(In this theorem, it is not necessary to ask thatF is closed under minors. We may aonly thatU0,1 belongs toF and so all the uniform matroids belong toF .)


rs

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dts have

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e

e

tt

Proof. Let Fk be the set of matroids on ak-element setAk belonging toF . We mayassume thatAm ∩ An = ∅ andAm+n = Am ∪ An. We define a functionΨ :Fm × Fn →Fm+n as follows. For matroidsM ∈Fm andN ∈ Fn such that

(D1) |E(M)| − | loop(M)| = 0 or |E(N)| − | loop(N∗)| = 0, we defineΨ (M,N) =M ⊕ N ;

(D2) |E(M)| − | loop(M)| = 1 and |E(N)| − | loop(N∗)| � 1, we defineΨ (M,N) =(M\e) ⊕ [N∗ + e]∗, wheree ∈ E(M) − loop(M);

(D3) |E(M)| − | loop(M)| � 2 and |E(N)| − | loop(N∗)| = 1, we defineΨ (M,N) =[M + e] ⊕ (N\e), wheree ∈ E(N) − loop(N∗);

(D4) |E(M)| − | loop(M)| � 2, |E(N)| − | loop(N∗)| � 2 and there are positive integer, s, andn and a matroidH satisfyings < n, whenH is non-empty, ors < r, whenH is empty, such that(M,N) is a(r, s, n,H)-pair, we defineΨ (M,N) = M �N ;

(D5) |E(M)| − | loop(M)| � 2, |E(N)| − | loop(N∗)| � 2 and there are positive integer, s and n such thats < n and a non-empty matroidH such that(N∗,M∗) is a(r, s, n,H)-pair, we defineΨ (M,N) = [N∗ �M∗]∗;

(D6) otherwise, we defineΨ (M,N) = M N .

Observe that:

(a) If Ψ (M,N) is defined in (D1), thenΨ (M,N) has at leastm loops orn coloops;(b) If Ψ (M,N) is defined in (D2), thenΨ (M,N) hasm − 1 loops and at mostn − 1

coloops;(c) If Ψ (M,N) is defined in (D3), thenΨ (M,N) has at mostm − 2 loops andn − 1

coloops;(d) If Ψ (M,N) is defined in (D4), (D5) or (D6), thenΨ (M,N) has at mostm − 2 loops

andn − 2 coloops;(e) If Ψ (M,N) is defined in (D4) or (D5), thenΨ (M,N) has exactly two connecte

components with at least two elements. Moreover, these connected componendifferent number of elements if and only ifH is non-empty; and

(f) If Ψ (M,N) is defined in (D6), thenΨ (M,N) has exactly one connected componwith at least two elements.

The result follows provided we show thatM ∼= M ′ and N ∼= N ′ if and only ifΨ (M,N) ∼= Ψ (M ′,N ′). Note that the “only if” part of this assertion is easy to verWe will prove the “if” part only. Suppose thatΨ (M,N) ∼= Ψ (M ′,N ′). If Ψ (M,N) has atleastm loops, then, by (a) to (d),Ψ (M,N) andΨ (M ′,N ′) are defined in (D1). HencM ⊕ N ∼= M ′ ⊕ N ′. If M ∼= M ′ ∼= U0,m, thenN ∼= N ′. If M ∼= U0,m and N ′ ∼= Un,n,then M ∼= M ′ and N ∼= N ′. We may assume thatΨ (M,N) has at mostm − 1 loops.Similarly, we may assume thatΨ (M,N) has at mostn − 1 coloops. IfΨ (M,N) hasm − 1 loops, then, by (a) to (d),Ψ (M,N) and Ψ (M ′,N ′) are defined in (D2). HencM ∼= M ′ ∼= U0,m−1 ⊕ U1,1. Moreover, ife ande′ are the coloops ofM andM ′ respec-tively, then(M\e)⊕[N∗ + e]∗ ∼= (M ′\e′)⊕[N ′ ∗ + e′]∗. SoN ∼= N ′. We may assume thaΨ (M,N) has at mostm − 2 loops. Similarly, we may assume thatΨ (M,N) has at mosn − 2 coloops.


ents,

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ents.m-then

tedeeve

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3)

is-

If Ψ (M,N) ∼= Ψ (M ′,N ′) has just one connected component with at least two elemthen, by (e) and (f),Ψ (M,N) andΨ (M ′,N ′) are defined in (D6). ThusM N ∼= M ′ N ′.By Lemma 4, whenM �∼= M ′ or N �∼= N ′, then

(g) there is a matroidH and positive integersr, s andn such that(M,N) is a(r, s, n,H)-pair and(M ′,N ′) is a(s, r, n,H)-pair; or

(h) there is a matroidH and positive integersr, s and n such that(N∗,M∗) is a(r, s, n,H)-pair and(N ′ ∗,M ′ ∗) is a(s, r, n,H)-pair.

If (g) happens, thenr < s or s < r. If s < r � n, then Ψ (M,N) should be definedin (D4); a contradiction. Ifr < s � n, then Ψ (M ′,N ′) should be defined in (D4)a contradiction. We have similar contradictions when (h) happens. We may assumΨ (M,N) ∼= Ψ (M ′,N ′) has exactly two connected components with at least two elemThat is,Ψ (M,N) andΨ (M ′,N ′) are defined in (D4) or (D5). If the two connected coponents ofΨ (M,N) ∼= Ψ (M ′,N ′) with more than two elements have the same size,Ψ (M,N) andΨ (M ′,N ′) are defined in (D4). ThusΨ (M,N) = M �N = M ⊕ N andΨ (M ′,N ′) = M ′ ⊕ N ′. HenceM ∼= M ′ andN ∼= N ′ because the rank of the conneccomponent with at least two elements ofM (or M ′) is not bigger than the rank of thconnected component with at least two elements ofN (or N ′). We may assume that thtwo connected components ofΨ (M,N) ∼= Ψ (M ′,N ′) having at least two elements hadifferent sizes. Using Lemma 6, its dual and Lemma 7, we conclude thatM ∼= M ′ andN ∼= N ′. The result follows. �

We say that a classF of matroids isclosed under Hamiltonian extensions provided, foreachN ∈F , there isM ∈F ande ∈ E(M) such thate belongs to a Hamiltonian circuit oM andN = M\e.

Conjecture 1. Let F be a class of matroids closed under duality, isomorphisms, minors,1-sums, 2-sums and Hamiltonian extensions. If m and n are non-negative integers, then

f (m + n) � f (m)f (n),

where f (k) denotes the number of non-isomorphic matroids on a k-element set belongingto F .

Observe that the classFk of matroids linearly representable over a fieldk satisfies thehypotheses of this conjecture. ForFGF(2), this conjecture was made by Wild [7].

References

[1] J.E. Blackburn, H.H. Crapo, D.A. Higgs, A catalogueof combinatorial geometries, Math. Comp. 27 (197155–166.

[2] W.H. Cunningham, A combinatorial decomposition theory, PhD thesis, University of Waterloo, 1973.[3] M. Lemos, J. Oxley, On the 3-connected matroids thatare minimal having a fixed spanning restriction, D

crete Math. 218 (2000) 131–165.


in. 24

ite

[4] M. Lemos, J. Oxley, On the minor-minimal 3-connected matroids having a fixed minor, Europ. J. Comb(2003) 1097–1123.

[5] J.G. Oxley, Matroid Theory, Oxford University Press, New York, 1992.[6] D.J.A. Welsh, A bound for the number of matroids, J. Combin. Theory 6 (1969) 313–316.[7] M. Wild, The asymptotic number of inequivalent binary codes and nonisomorphic binary matroids, Fin

Fields Appl. 6 (2000) 192–202.

On the number of non-isomorphic matroids

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