AbstractThe purpose of this note is to point out an error in the paper [2], named in the title above, published under my authorship in this Journal in 1971

O'Neil, Patrick E.

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Correction to a constructive decomposition and Fulkerson's characterization of permutation matrices 

Discrete Mathematics 11 ( 1975) 401-402. 
0 North-Holland Publishing Company 
NOTE 
CORRECTION TO A CONSTRUCTIVE DECOM OSHWN AND 
FWLKERSON’S C ACI’ERIZATION OF PERMUT 
MATRICES 
Kecx&ed 23 August 1974 
The purpose of this note is to point out an ert~r in the paper 12 j, nsmed in the title 
above, published under my authorslAp in this Journal in 1971. 
Consider the set P of H >G 11 matrices X = (-Yij) for which 
for all I. J Ir, 11, 2, ,..* fj}, with xii 2 0 for ail i, j f { 1, 2, ,.., n). 
III [ 21, I presented what I t’loirght was an elernentarv constructive c 
proof of the theorem: 
The result is true and had paeviously been proven by Fulkerson 
The proof in [I’?], however, is invalid. 
In [I!], the following lemma was stated with adequate proof: 
El1 . 
The invalid proof that if X E P then X = S + N runs as fsliisws. If no 
S is greater than I, t is doubly stoctiastic. 
X has a row i an c~~u~~n j as specified in 
Xij rwv swn i = 1, or (b) c 
9% I? E. O’Neii / Permutartiurr matrices 
This gives 3 new matrix X Q1) Continuing in this manner, me note that in . 
finite time 2 matrix Xck) will result, where no row sums are greater than 
ne, ‘Thus X = kytk) + IV, where Xck) is doubly stochastic and N has entries 
~~rr~~~onding to the non-negative ntries subtracted from X. 
231 E 073 : received a ktter fmm .Ailm ES, Cruse, University of San 
rancisco~ pointing out that the in&lcticn hypothesis required that each 
uc~~ssive JI?fi be in P, but that this was not proved. In fact it is not true, 
8s the following counterexample shows. Consider the matrix, 
It is easily seen that the matrix M iieb in P since it contains the permu- 
t n matrix 
ate too that: row sum 1 =2> 1 andcolumnsum3=2> l.Butif 
ntry M( 1.3) is reduced to zero (as the proof above would have it), 
suft is not in P and contains no doubly stochastfc matrix: to see 
t/= (I,3) andJ= {1,3). 
I have had no success in finding a correct elementary constructive 
A.B. Cruse has produced a generalization of this theorem which 
dmirabie. P hope it will soon be published. My own paper retains 
c va!ue since it contains a proof that a subset of the inequalities de- 
are “essential”, as asserted by Fulkerson in [ 11 without proof. 
n. Hocking Polyhedra, Graph Theory and itc -4ppiications (Academic Press, 
xl). 
construciive decomgc %CXI iand Fulkerson’s characterizaticw of permutation 
screte Math. 1 42) (1971) 197-201. 


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Correction to a constructive decomposition and Fulkerson's characterization of permutation matrices

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