On the DJL conjecture for order 6

Abstract

In 1994 Drew, Johnson and Loewy conjectured that for $n\ge 4$, the cp-rank of any $n\times n$ completely positive matrices is at most $\lfloor n^2/4\rfloor$. Recently this conjecture has been proved for $n=5$ and disproved for $n\ge 7$, leaving the case $n=6$ open. We make a step toward proving the conjecture for $n=6$. We show that if $A$ is a $6\times 6$ completely positive matrix that is orthogonal to an exceptional extremal copositive matrix, then the cp-rank of $A$ is at most $9$.

16 pages, 1 table, 11 figures. This version contains corrections (shown in blue): to the proof Lemma 3.1, and to Lemma 2.11. The error was detected and corrected by Peter J. C. Dickinson

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• Extremal copositive matrices with minimal zero supports of cardinality two