CHAPTER 2

CHARACTERIZATIONS OF BINARY

EULERIAN *

^ A paper based on the text included in this chapter has been publshed in Indian J. Pure Appl. Math., see Reference No.13 isted at the end. 2.1 INTRODUCTION

Eulerian and bipartite matroids have been discussed by Welsh [16]. We recall that a graph is Eulerian if it contains a closed walk that traverses each edge exactly once. It is well known that a graph is Eulerian if and only if its edge set can be partitioned into cycles. Analogously a (S, ^) is Eulerian if there exist disjoint circuits C ,C ,...,C such that 12 ri S = C uC u ...uC 12 n

Also a graph is bipartite iff every cycle of it is of even

length. Analogously, a matroid (S, 5=") is defined to be bipartite

if every circuit has even cardinality.

Welsh [16] generalized Euler's well known theorem [see Theorem 1.8] for graphs to binary matroids. In this chapter we generalize Shank's characterization [Theorem 1.11] of Eulerian graphs to binary matroids [Theorem 2.3.1]. The Theorem 2.4.1 extends the characterization of Eulerian graphs [Theorem 1.10] due to Toida and Mckee whereas the Theorem 2.5.1 generalizes results [Theorem 1.13] due to Bondy and Halberstan for graphs. Corresponding results for binary bipartite matroids have been derived. The motivation for these results are from graph theory. They are either analogous to or generalizations of corresponding graph-theoretic results.

13 2.2. EULERIAN AND BIPARTITE MATROIDS

We begin with the theorem due to Welsh [16]. He observe that

for binary matroids being Eulerian and bipartite are dual

concepts. The proof of 'if part' presented here is different from

Welsh's proof.

THEOREM 2.2.1. A is Eulerian if and only if its

is bipartite.

PROOF : Let M = (S, ^) be Eulerian and binary, so that

S = C uC u ...uC 12 r.

where {C > are disjoint circuits of M. Let D be any circuit of M (that is, any cutset of M). Then by Theorem 1.32,

I D n C, I = 2k , 1 < i < n

n n ^ hence |D|=EtDnC. [=2 E k.,an even number. Thus M is i. =1 i =1 bipartite.

Suppose now that M is binary and bipartite. If C is one of its cut sets then M .p is also binary by Theorem 1.43, and is

bipartite by Proposition 1.48. If C^ is a cutset of this latter

then (M.p ) .„ is bipartite again and so on. Now C is a circuit 12 in (M .^ ) . And by proposition 1.40, (M^^ ) = M. „ - M. ^ so that

C^ is a circuit in M. ^ as well as a circuit in M. Thus, S is

partitioned into circuits of M as C^ u C, u ... and hence

M = (S.^y^) is an .

14 REMARK 2.2.2

The above theorem does not hold for non-binary matroida. Let

S be any set of cardinality 2k+l (k odd and > 3). Let ^ consists of all sets with cardinality ^ k. Then the circuits of M = (S, ^) are all sets with cardinality k+1, an even number. Thus M is bipartite but M = {S, •?" ) is not Eulerian. Since ^ consists of

all subsets of S with cardinality ^ k+1, the circuits of M will have cardinaity k+2. And |S| = 2k+l ^ m (k+2) for some m e IN.

COROLLARY 2.2.3 Let G be any graph and let (S,5^) be the circuit matroid of G. Then (S,5^) is an Eulerian matroid iff G is Eulerian. Thus the cutset matroid of an Eulerian graph must be a bipartite matroid and this gives a matroid proof of the following well known results in graph theory.

(1) The faces of a plamar graph G are 2-colourable iff G is an Eulerian graph. (2) G is an Eulerian graph iff each of its cutsets has even cardinality.

(3) G is a iff its edge set can be partitioned into disjoint cutsets.

2.3. EULERIAN MATROIDS AND INDEPENDENT SETS

In the following theorem we provide necessary and sufficient

conditions in terms of number of independent sets for a binary matroid to be Eulerian. This generalizes Theorem 1.11 for graphs due to Shank [12].

15 THEOREM 2.3.1 : A binary matroid M on a set S is Eulerian if and only if the n\anber of independent sets of M is odd,

PROOF : Let M = (8,5^) be a binary matroid. Let r^ be the number of independent sets of M and ^^(x) be the corresponding number of independent sets of M containing x. Since a loop is a dependent set of M, we can assume without loss of generality that M is loopless. We proceed by induction on |S|. If |S| =2 and M is Eulerian then a base of M consists of singleton subsets of S and hence trivially the number of independent sets of M is odd.

Let now jS| > 2 and x be an arbitrary element of S. Form the

matroid M., , by contracting x in M, and matroid M. , , by deleting /{x} \lx> X from M. By Theorem 1.43, both M,, , and M. , ^ aj^e binary matroids.

The independent sets of M are divided into two classes. The first class consists of the independent sets that include x and the second class those not containing x. An independent set in the first class, say X corresponds to the independent set X' = X--Cx} of M-r -, and an independent set X' of M., , corresponds /Ix} /{x> '^ to the independent set X = X' u {x> of M in the first class. Also we note the one-one correspondence between independent sets of M containing x and independent sets of M ., ,. So by induction, r : r^ = r^Cx) (1) M M

with r = 1 (mod 2) if and only if H ., -, is Eulerian. . . . (2) /lx> Further, if Y is an independent set of M in the second class, then Y is an independent set of M. , ,. So, we conclude from the \ix> above that 16 (3) r - Y + r M M M /{x} \{x}

Therefore if M is Eulerian then by Lemma 1.44., \{.r-i is Eulerian. However M. ^ , is not Eulerian. By induction and by (2) \lx> and (3)

r =1+0=1 (mod 2);

I.e. r^ s 1 (Mod 2).

This proves the "only if part.

For the "if part' assume that M = (Sj-y) is not an Eulerian matroid. If for some x e S neither of M., , and M. ^ is Eulerian then by induction and by (2) and (3) we have

r^^ s 0 + 0 = 0 (mod 2).

If however, M. , -, is Eulerian then by LenHna 1.44, M.^ , IS Xlx} /tx> Eulerian. Similarly, if M ., , is Eulerian then M. , , is Eulerian. /{x} \lx} Thus, if one of M,, , and M. ^ -> is Eulerian then both are /•[x} \ix} Eulerian.

(k)nsequent ly,

r^ = 1+1 = 0 (mod 2) i.e. r = 0 (mod 2) if M is not Eulerian. This completes the proof of the theorem.

REMARK 2.3.2 The above result does not hold good for non-binary matroids as shown by the following example.

17 Let U be a of rank 2 on a six element set.

This is a non binary Eulerian matroid. Then the number of independent sets in U is 22, an even niomber. On the other hand «5,2 the number of independent sets in U is 11 but U is not

Eulerian.

In general let U be the uniform matroid of rank 2 on an n ,2 n-element set (this is binary for n > 3). One can see that it is Eulerian if and only if n is conginient to 0 (mod 3) and the number of its independent sets is odd if and only if n is congruent to 0 or 3 (mod 4). Since 3 and 4 are relatively prime nvunbers any combination of the two properties can arise.

The following corollary gives a characterization of binary bipartite matroids. The proof follows from Theorems 2.2.1 and 2.3.1.

COROLLARY 2.3.3 : A binary matroid M is bipartite if and only if the numebr of independent sets of M is odd. a. 4. EULERIAN MATROIDS AND CIRCUITS :

Following theorem extends to binary matroids the Theorem 1.10. It provides a characterization of Eulerian binary matroids in terms of circuits.

THEOREM 2,4.1 : A binary matroid M = (S, 3^) is Eulerian if and only if every element of S is contained in an odd number of circuits of M.

18 PROOF : Suppose a binary matroid M = (S,3^) is Eulerian so every cutset of M has even cardinality. Let x e S. We show that x is contained in an odd number of circuits of M. Let S^ denote the set of circuits containing x and A«^ denote the symmetric difference of members of « . Let C(x ,x, ,...,x ) denote the number of X 1 Z n circuits containing x ,x ,...,x^- Then prove that

(i) R = A'e u {x} intersects each cutset in even number of X X elements. Let D be a cutset of M. If x «s D, then D n R = D n {A« u {x}) = D n AIS is even by binarity of M. Now, XX X let x e D and D n R has odd number of elements. Suppose D = {x,x ,...,x }. Then n is odd. (D n R )-x is even and n E C(x,x ) is also even. Therefore at least one of the n tenns i. =1 must be even. Let C(x,x ) be even and let D be any cutset containing x but not x,. Say D = {x,y ,y , . . . ,y ->. Now 1 1 12 n n' C(x,x^) = E C(x,x^,y ) is even, therefore at least one of the n' terms must be even; say C(x,x ,y ) is even. Let D be any cutset containing x but neither x^ nor y etc. Continue this process until cutset D can not be chosen. Then the i elements x ,y , . . . . together with x form a circuit and so C(x,x ,y ,. . .) = 1. But we also have C(x,x ,y , . . .) even and this is the contradiction. Therefore D n R must have odd number of elements. X (ii) By proposition 1.49, R is a disjoint union of circuits of

M. Also, AT? is a disjoint union of circuits and R = AS u {x}. X XX So, we must have x e A15 . Consequently the number of circuits containing x must be odd. 19 Conversely, suppose that a matroid M = (S.-y) is binary and every element is contained in odd number of circuits. In order to prove that M is Eulerian, we prove that every cutset of M is of even size.

Let D be any cutset of M and C{x) denote number of circuits containing the element x. Since M is binary by Theorem 1.32, every circuit of M intersects D evenly, the sum E C(x) over all x e D will count each circuit an even number of times. So, the sum

D C(x) will be even. By assumption each term is odd, so there X € D must be an even number of terms, thereby showing D to be even. Consequently, M is Eulerian. REMARK 2.4.2 This result is not true for non binary matroids. Consider the matroid LJ of Remark 2.3.2. U is non binary

20 COROLLARY 2.4.3. A binary matroid M = (S,^) is Eulerian if and only if synmetric difference of all circuits of M equals S.

Following corollary to the above theorem gives corresponding characterization for bipartite matroids.

COROLLARY 2.4.4. A binary matroid M = (S,-?') is bipartite if and only if every element is contained in an odd number of cutsets of

M*.

2.5. EULERIAN MATROIDS AND PARTITIONS

Following theorem gives a characterization of binary Ehilerian matroids in terms of partitions of underlying set into circuits. This generalizes the characterization [Theorem 1.13] of Eulerian graphs due to Bondy and Halberstan to the binary matroids.

THEOREM 2.5.1 A binary matroid M = (S, ^) is Eulerian if and only if S ccun be partitioned into circuits of M in an odd number of ways.

PROOF If in a binary matroid M the underlying set S can be

partitioned into circuits of M in an odd number of ways then it surely has at least one circuit partition, therefore M= (S,-?") is Eulerian.

Now suppose M = (S,-?') is Eulerian and binary. Let x « S and C ,C , - . . , C, be the circuits containing x. Then by Theorem 2.4.1, k = 1 (mod 2). We proceed by induction on jSJ. If |Sj = 1 then S has the trivial partition consisting of a loop.

21 r^t I si = n > 1. If S. = S-C =

as its unique circuit partition. Assume therefore that S ^

M. o = (S-C , ^ ) has an odd number of circuit partitions. This \L,. 1. ). x. yields an odd number of circuit partitions of M contaiing C .

Denote this number by T(C. ) and denote by T(M) the number of circuit partitions of M.

Consequently,

k T{M) = E T(C ) = k = 1 (mod 2)

i.e. T(M) H 1 (mod 2).

REMARK 2. 5. 2 This also does not hold for non-binary matroids as shown by the following example.

(Consider the matroid U [see Remark 2.3.2]. Every 3-element set is a circuit. Thus, there are C =20, circuits. Now any circuit and its complement in the 6-element set which is also a circuit form the partition of S into circuits of U . Thus in all

Theorem 2.5,1 enables us to give another characterization of binary bipartite matroids in terms of partitions of underlying set into cutsets.

COROLLARY 2.5.2 A binary matroid M = (S,5^) is bipartite if and only if S can be partitioned into cutsets of M in odd number of ways.

22 2.6. ALGORITHM FOR CIRCUIT PARTITIONS

Here we present a description of an algorithm for the construction of the set ^(S), of all circuit partitions of S in a binary Eulerian matroid M = {S,-5^)- Every Eulerian matroid has at least one circuit, C(say) and M.p = (S-C ^') is also Eulerian. We use this to describe an algorithm to obtain circuit partitions of arbitrary binary Eulerian matroid.

Suppose M = (S,.^) is a Eulerian matroid and

S = tx ,x ,...,x >. Choose an arbitrary element x e S, x = x 1 Z n tl say. A first subroutine produces in lexicographic order the set "6 of all circuits containing x. For C e « , the matroid M.p = (S-C , ^') is Eulerian. A second subroutine produces the i set ^. of all circuit partitions of S in the matroid M which have

C. in common. This is achieved by determining 5*(S-C. ) and forming

jp = {{C } up I p € ^ (S-C)} t \. Then we note that :P. n J>. = cp for i?^j, l

Now, it follows that r X P(.S) = \J P i. 1=1

is the required set. Here each ^. is a partition of S into circuits of M.

23 REMARK 2.6.1. (1) We have |^(S)| =1 for a matroid M each of whose component is a circuit.

(2) In Theorem 2.5.1, the number of partitions of S into circuits of M is odd if and only if M = (8,5=") is binary Eulerian matro id. Hence

l^

24