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Chapter 2 Characterizations of Binary Eulerian Matroids*

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Chapter 2 Characterizations of Binary Eulerian Matroids* CHAPTER 2 CHARACTERIZATIONS OF BINARY EULERIAN MATROIDS* ^ 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 matroid (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 binary matroid is Eulerian if and only if its dual matroid 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 Eulerian matroid. 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 bipartite graph 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 uniform matroid 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.
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