OJDM  Vol.7 No.3 , July 2017
On Functions of K-Balanced Matroids
Author(s) Talal Al-Hawary
ABSTRACT
In this paper, we prove an analogous to a result of Erdös and Rényi and of Kelly and Oxley. We also show that there are several properties of k-balanced matroids for which there exists a threshold function.

1. Introduction

We begin with some background material, which follows the terminology and notation in [1] . Let M = ( E , F ) denote the matroid on the ground set E with flats F. All matroids considered in this paper are loopless. In particular, if M is a matroid on a set E and X ⊆ E, then r(X) will denote the rank of X in M. We shall be considering projective geometries over a fixed finite field GF(q), recalling that

(see, for example [2] ) the number [ r n ] of rank-n subspaces of the projective

geometry PG (r − 1, q) is

( q r 1 ) ( q r 1 1 ) ( q r n + 1 1 ) ( q n 1 ) ( q n 1 1 ) ( q 1 ) .

The uniform matroid of rank r and size n is denoted by U r , n where

r = 0 , 1 , , n . When r = n, the matroid U r , r is called free and when r = n = 0, the matroid U 0 , 0 is called the empty matroid. For more on matroid theory, the reader is referred to [1] - [15] . Let k be a nonnegative integer. The k-density of a

matroid M with rank greater than k is given by d k ( M ) = | M | r ( M ) k , where |M|

is the size of the ground set of M and r(M) is the rank of the matroid M. A matroid M is k-balanced if r ( M ) > ( k ( k + 1 ) ) / 2 and

d k ( M ) d k ( M ) (1)

for all non-empty submatroids H M and strictly k-balanced if the inequality is strict for all such H ≠ M. When k = 0, M is called balanced and when k = 1, M is called strongly balanced.

A random submatroid ω r of the projective geometry P G ( r 1 , q ) is obtained from P G ( r 1 , q ) by deleting elements so that each element has, independently of all other elements, probability 1 − p of being deleted and probability 1 − p of being retained. In this paper, we take p to be a function p(r) of r. Let A be a fixed property which a matroid may or may not possess and P r , p ( A ) denotes the probability that ω r has property A. We shall show that there are several properties A of k-balanced matroids for which there exists a function t(r) such that

lim r P r , p ( A ) = { 0 , lim r P t ( r ) = 0 1 , lim r P t ( r ) =

If such a function exists, it is called a threshold function for the property A. For more on these notions, the reader is referred [16] [17] .

2. K-Balanced Matroids

In this section, we prove the following main result which is analogous to Theorem 1 of Erdös and Rényi [16] and to Theorem 1.1 of Kelly and Oxley [17] .

Theorem 1. Let m and n be fixed positive integers with n ≤ m and suppose that B n , m denote a non-empty set of k-balanced simple matroids, each of which have m elements and rank n and is representable over GF(q). Then a threshold function for the property B that ω r has a submatroid isomorphic to some

member of B n , m is q r n m .

Proof. Let X and B n , m denote the number of submatroids of the matroid ω r and P G ( n 1 , q ) respectively which are isomorphic to some member of B n , m . Then

P r , p ( B ) = P ( X 0 ) E X

by definition of expectation. Therefore

P r , p ( B ) [ r n ] B n , m p m B n , m p m q r n B n , m ( p q r n m ) m .

Thus, if lim r p q r n m = 0 , then lim r P r , p ( B ) = 0 .

Now suppose that lim n p q r n m = . We need to show that, in this case, lim n P r , p ( B ) = 1 . Let D m , n be the set of subsets A of P G ( r 1 , q ) for which the

restriction P G ( r 1 , q ) | A of P G ( r 1 , q ) to A is isomorphic to some member of B n , m . Then

E X 2 = A 1 D m , n A 2 D m , n p | A 1 A 2 | = i = 0 m p m + i i (2)

where i equals the number of ordered pairs ( A 1 , A 2 ) such that A 1 , A 2 D m , n and | A 1 A 2 | = m i . Thus

E X 2 p 2 m [ ( B m , n [ r n ] ) 2 + i = 0 m 1 p i m i ] .

We now want to obtain upper bounds on the numbers 0 , 1 , , m 1 , so suppose that A 1 , A 2 D m , n and | A 1 A 2 | = m i where 0 i m 1 . Then as P G ( r 1 , q ) | A is k-balanced,

( | A 1 A 2 | ) / ( r ( A 1 A 2 ) k ) m / ( n k )

and so r ( A 1 A 2 ) ( ( m i ) ( n k ) ) / m + k . It follows that

r ( A 2 ) r ( A 1 A 2 ) n ( ( m i ) ( n k ) ) / m k = ( i ( n k ) ) / m ( i n ) / m

and hence r ( A 2 ) r ( A 1 A 2 ) ( i n ) / m where ( i n ) / m is the floor of ( i n ) / m .

Now i = β i γ i where β i is the number of ways to choose A 1 and γ i is the number of ways to choose A 2 so that | A 1 A 2 | = m i , A 1 having already

been chosen. Clearly β i = B m , n [ r n ] . Once A 1 has been chosen, there are at

most ( m i m ) choices for the subset A 1 A 2 of A 1 . Further, once A 1 A 2 has been chosen, A 2 must be contained in some rank n subspace W of PG(r-1,q) which contain the chosen set A 1 A 2 . The number δ of such subspaces W is bounded above by

( ( q r q s ) / ( q 1 ) ) ( ( q r q s + 1 ) / ( q 1 ) ) ( ( q r q n 1 ) / ( q 1 ) ) ,

where s = r ( A 1 A 2 ) . Thus δ q r ( n 1 ) . But it was shown above that

n s ( i n ) / m ; hence δ q r i n / m . Once W has been chosen, there are at most B m , n choices for A 2 . We conclude that

γ i ( m i m ) q r i n / m B m , n

and hence

α i [ r n ] B m , n 2 ( m i m ) q r i n / m . (3)

Now as E X = [ r n ] B m , n p m , we have by Equation (2), that

E X 2 ( E X ) 2 1 + ( B m , n [ r n ] ) 2 + i = 0 m 1 p i m i .

Hence, by Equation (2),

E X 2 ( E X ) 2 1 + ( B m , n [ r n ] ) 2 + i = 0 m 1 p i m [ r n ] B m , n 2 ( m i m ) q r i n m . .

Thus E X 2 ( E X ) 2 1 + i = 0 m 1 p i m ( m i m ) q r i n m [ r n ] 1 + i = 0 m 1 p i m ( m i m ) q r i n m q n ( r n )

Since [ r n ] q n ( r n ) . Thus

E X 2 ( E X ) 2 1 + i = 0 m 1 p i m q r n + r i n m ( m i m ) q n 2 . (4)

Now consider p i m q r n + r i n m . We have

q r n + r i n m q r ( n i n m ) = ( q r n / m ) i m .

Thus p i m q r n + r i n m ( p q r n m ) i m . But lim r p q r n m = , hence lim r ( p q r n / m ) i m = 0 for 0 i m 1 . It follows from Equation (4) that lim r sup E X 2 ( E X ) 2 1 ; hence lim r E X 2 ( E X ) 2 = 1 . Therefore, by Chebyshev’s Inequality, lim r P ( X 0 ) = 1 . We conclude that q r n m is indeed a threshold function for the property B.

Corollary 1 If n is a fixed positive integer, then a threshold function for the property that ω r has an n-element independent set is q r .

Corollary 2 If m is a fixed positive integer exceeding two, then a threshold

function for the property that ω r has an m-element circuit is q r ( m 1 ) m .

Corollary 3 If n is a fixed positive integer, then a threshold function for the property that ω r contains a submatroid isomorphic to P G ( n 1 , q ) is

q r n ( q 1 ) q n 1 .

To show that the preceding three results are valid, we are required to check that the appropriate submatroids are k-balanced. For example, in Corollary 1, the n-element independent set must be k-balanced; this is the free matroid U n , n . Corollary 2 requires one to verify that an m-element circuit is k-balanced; this is precisely the uniform matroid U m 1 , m , while in Corollary 3, the projective geometry P G ( n 1 , q ) needs to be k-balanced. For a more thorough discussion of this material, the reader is referred to Proposition 2 and Theorem 5 in [2] .

Cite this paper
Al-Hawary, T. (2017) On Functions of K-Balanced Matroids. Open Journal of Discrete Mathematics, 7, 103-107. doi: 10.4236/ojdm.2017.73011.
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