Packing Tree Degree Sequences

Abstract

A degree sequence is a list of non-negative integers, {D = d_1, d_2, ldots , d_n}. It is called graphical if there exists a simple graph G such that the degree of the ith vertex is d_i; G is then said to be a realization of D. A tree degree sequence is one that is realized by a tree. In this paper we consider the problem of packing tree degree sequences: given k tree degree sequences, do they have simultaneous (i.e. on the same vertices) edge-disjoint realizations? We conjecture that this is true for any arbitrary number of tree degree sequences whenever they share no common leaves (degree-1 vertices). This conjecture is inspired by work of Kundu (SIAM J Appl Math 28:290–302, 1975) that showed it to be true for 2 and 3 tree degree sequences. In this paper, we give a proof for 4 tree degree sequences and a computer-aided proof for 5 tree degree sequences. We also make progress towards proving our conjecture for arbitrary k. We prove that k tree degree sequences without common leaves and at least 2k-4 vertices which are not leaves in any of the trees always have edge-disjoint tree realizations. Additionally, we show that to prove the conjecture, it suffices to prove it for n le 4k - 2 vertices. The main ingredient in all of the presented proofs is to find rainbow matchings in certain configurations.