Packing of graphic n‐tuples

Abstract

An n-tuple π (not necessarily monotone) is graphic if there is a simple graph G with vertex set {v 1 , . . . , v n } in which the degree of v i is the ith entry of π. Graphic n-tuples (d (1) 1 , . . . , d(1)for all i. We prove that graphic n-tuples π 1 and π 2 pack if ∆ ≤ √ 2δn − (δ − 1), where ∆ and δ denote the largest and smallest entries in π 1 + π 2 (strict inequality when δ = 1); also, the bound is sharp.Kundu and Lovász independently proved that a graphic n-tuple π is realized by a graph with a k-factor if the n-tuple obtained by subtracting k from each entry of π is graphic; for even n we conjecture that in fact some realization has k edge-disjoint 1-factors. We prove the conjecture in the case where the largest entry of π is at most n/2 + 1 and also when k ≤ 3.

Graphic Sequences Have Realizations Containing Bisections of Large Degree

“…One of |A∩X| and |A∩Y| is at most p / 2 since |A| = p. Without loss of generality, assume |A∩Y| ≤ p / 2. Any vertex in C ∩X has at most p / 2 neighbors in A∩Y, p neighbors in B, and no neighbors in C. Therefore, it has at most 3 2 p neighbors in Y. Case 2: C is completely contained in X or Y.…”

“…We claim H is a 3 2 p-regular bipartite graph with partition X=A 1 ∪B 1 ∪C 1 and Y=A 2 ∪B 2 ∪C 2 . We check that every vertex in X has degree 3 2 p, and the vertices in Y will have the same degree by symmetry. The vertices in C 1 have p / 2 edges to A 2 and p edges to B 2 , and therefore vertices in C 1 have degree 3 2 p. Vertices in A 1 have p / 2 edges to C 2 and p edges to B, and therefore have degree 3 2 p. Finally, the vertices in B 1 have one edge to A 2 , one edge to C 2 , and 3 2 p−2 edges to B 2 .…”

Packing Tree Degree Sequences

“…In the inductive step, we need the following lemma to reduce the case to a case with fewer number of vertices. (2) , . .…”

Forced Edges and Graph Structure

“…The second term uses a degree sequence packing result. Using the observation that two degree sequences cannot pack where both have the same forced edge v 1 v 2 , we apply Theorem 2.2 of Busch et al [3] to pack a sequence α with the sequence (1, 1, 0, ..., 0) and after some algebraic manipulation establish the second bound.…”