Abstract :
LetA be a doubly stochastic matrix of ordern andσ1(A) the sum of the orderi subpermanents ofA. The Holens-Đoković Conjecture says inσ1(A) greater-or-equal, slanted (n−i+1)2σi−1A) for eachi = 1,2…n. There is a natural counterpart of this conjecture for (0, 1)-matrices with constant line sumk. We show this associated conjecture holds whenk less-than-or-equals, slant 2, k greater-or-equal, slanted n − 2, i less-than-or-equals, slant n/k + 1 or i less-than-or-equals, slant 8 but that it fails in general because it (wrongly) asserts a polynomial bound on the ratio of near perfect to perfect matchings ink-regular bipartite graphs.
