Polynomials with coefficients in the sets \(\{-1,1\}\), \(\{0,1\}\) and \(\{-1,0,1\}\) are called Littlewood-Newman-Borwein polynomials, respectively. In [Math. Comput. 78, No. 265, 327--344 (2009; Zbl 1208.11123)], the reviewer and the second author investigated various divisibility relations between these three sets of polynomials. In particular, they showed that every Newman polynomial of degree at most \(8\) divides some Littlewood polynomial. In this paper, the authors show that every Borwein polynomial of degree at most \(8\) which divides some Newman polynomial also divides some Littlewood polynomial (Theorem 2). They also show that the least degree Borwein polynomial which does not divide any Littlewood polynomial is \(p(x)=x^4+x^3-x-1\) (Proposition 3). Three other polynomials with the same property are \(-p(x), \pm p^*(x)\). The number of such polynomials increases rapidly with degree. For instance, there are exactly \(16084\) Borwein polynomials of degree \(9\) which have no Littlewood multiple. In the above mentioned earlier paper, it was shown that each Newman polynomial of degree at most \(8\) has a Littlewood multiple, but there are degree \(9\) polynomials which do not have. Now, the authors give a complete list of \(18\) Newman polynomials which do not have a Littlewood multiple. They also show that there are exactly \(36\) such polynomials of degree \(10\) and exactly \(174\) such polynomials of degree \(11\).