A note on packing of graphic nn-tuples

A nonnegative integer nn-tuple (d1,…,dn)(d1,…,dn) (not necessarily monotone) is graphic   if there is a simple graph GG with the vertex set {v1,…,vn}{v1,…,vn} in which the degree of vivi is didi. Graphic nn-tuples (d1(1),…,dn(1)) and (d1(2),…,dn(2))pack   if there are edge-disjoint nn-vertex graphs G1G1 and G2G2 with the same vertex set {v1,…,vn}{v1,…,vn} such that dG1(vi)=di(1) and dG2(vi)=di(2) for all ii. Let Δ(π)Δ(π) and δ(π)δ(π) denote the largest and smallest entries in nn-tuple ππ respectively. For graphic nn-tuples π1π1 and π2π2, Busch et al. (2012) proved that if Δ(π1+π2)≤2δ(π1+π2)n−(δ(π1+π2)−1) (strict inequality when δ(π1+π2)=1δ(π1+π2)=1), then π1π1 and π2π2 pack. As a more direct analogue to the Sauer–Spencer Theorem, Busch et al. conjectured that if δ(π1+π2)≥1δ(π1+π2)≥1 and the product of corresponding entries in π1π1 and π2π2 is always less than n2, then π1π1 and π2π2 pack. In this paper, we prove that if δ(π1)≥1δ(π1)≥1, π2π2 is an almost kk-regular graphic nn-tuple with k≥1k≥1, and the product of corresponding entries in π1π1 and π2π2 is always less than n2, then π1π1 and π2π2 pack. This is an interesting variation of Kundu’s Theorem. Combining this result with a counterexample to the above conjecture of Busch et al., we present a slight modification of this conjecture as follows: if δ(π1)≥1δ(π1)≥1, δ(π2)≥1δ(π2)≥1 and the product of corresponding entries in π1π1 and π2π2 is always less than n2, then π1π1 and π2π2 pack.