Navigating between packings of graphic sequences

Let pi(1) = (d(1)((1)), . . ., d(n)((1))), and pi(2) = (d(1)((2)), . . ., d(n)((2))) be graphic sequences. We say they pack if there exist edge-disjoint realizations G(1) and G(2) of pi(1) and pi(2), respectively, on vertex set {v(1), . . ., v(n)} such that for j is an element of{1, 2}, d(Gj) (v(i)) = d(i)((j)) for all i is an element of (1, . . ., n). In this case, we say that (G(1), G(2)) is a (pi(1), pi(2))-packing. A clear necessary condition for graphic sequences pi(1) and pi(2) to pack is that pi(1) + pi(2), their componentwise sum, is also graphic. It is known, however, that this condition is not sufficient, and furthermore that the general problem of determining if two sequences pack is NP-complete. S. Kundu proved in 1973 that if pi(2) is almost regular, that is each element is from {k - 1, k}, then pi(1) and pi(2) pack if and only if pi(1) + pi(2) is graphic. In this paper we will consider graphic sequences pi with the property that pi + 1 is graphic. By Kundu's theorem, the sequences pi and 1 pack, and there exist edge-disjoint realizations G and I, where I is a 1-factor. We call such a (pi, 1) packing a Kundu realization. Assume that pi is a graphic sequence, in which each term is at most n/24, that packs with 1. This paper contains two results. On one hand, any two Kundu realizations of the degree sequence pi + 1 can be transformed into each other through a sequence of other Kundu realizations by swap operations. On the other hand, the same conditions ensure that any particular 1-factor can be part of a Kundu realization of pi + 1. (C) 2018 Elsevier B.V. All rights reserved.