On Erd˝os-Gallai and Havel-Hakimi algorithms

28 A. Iványi, L. Lucz, T. F. Móri, P. Sótér
Lemma 17 Let b be a random n-regular sequence. Let us write it in the form
b = (b e1
1
, . . . , ber
r ). Then the expected value of the exponents ej is
E[ej | r(b) ≥ j] = 4 + o(1). (47)
Proof. Let c(n, j) denote the number of n-regular sequences with rainbow
number not less than j. By Lemma 15,
n
n n − 1
c(n, j) =
. (48)
k k − 1
k=j
k=j
Let us turn to the number of n-regular sequences with rainbow number not
less than j and ej = ℓ. This is equal to the number of (0, n−1, n−ℓ+1)-regular
sequences containing at least j different numbers, that is,
n
n n − ℓ
. (49)
k k − 1
From this the sum of ej over all n-regular sequences with ej > 0 is equal to
n−j+1
ℓ=1
ℓ
n
k=j
n n − ℓ
k k − 1
=
n
n
k
k=j
=
n−j+1
n
k=j
ℓ n − ℓ
1 k − 1
ℓ=1
n n + 1
= c(n + 1, j + 1). (50)
k k + 1
This can also be seen in a more direct way. Consider an arbitrary n-regular
sequence with at least j + 1 blocks, then substitute the elements of the j + 1st
block with the number in the jth block (that is, concatenate this two adjacent
blocks) and delete one element from the united block; finally, decrease by 1
all elements in the subsequent blocks. In this way one obtains an n-regular
sequence with at least j blocks, and it easy to see that every such sequence is
obtained exactly ej times.
Thus the expectation to be computed is just
c(n + 1, j + 1)
.
c(n, j)
2n − 1
Clearly, c(n, 1) = R(0, n − 1, n) = , hence
n
(51)
2n − 1 j−1
n n − 1 2n − 1
c(n, j) = −
= + O n
n
k k − 1 n
2j−3
, (52)
k=1