Adjacency Relationships Forced by a Degree Sequence

Abstract

There are typically several nonisomorphic graphs having a given degree sequence, and for any two degree sequence terms it is often possible to find a realization in which the corresponding vertices are adjacent and one in which they are not. We provide necessary and sufficient conditions for two vertices to be adjacent (or nonadjacent) in every realization of the degree sequence. These conditions generalize degree sequence and structural characterizations of the threshold graphs, in which every adjacency relationship is forcibly determined by the degree sequence. We further show that degree sequences for which adjacency relationships are forced form an upward-closed set in the dominance order on graphic partitions of an even integer. arXiv:1508.00045v1 [math.CO] 31 Jul 2015

The principal Erdős--Gallai differences of a degree sequence

“…In [2,11] the author and Cloteaux independently studied forcible adjacency relationships such as these. Such a phenomenon is not restricted to split graphs like P 4 or to graphs with a nontrivial canonical decomposition; for example, in each of the nine labeled realizations of d = (4, 4, 3, 3, 3, 1), the vertices of degree 4 are adjacent, though the Erdős-Gallai difference list ∆(d) = (1, 1, 2, 2) contains no 0 term.…”

“…Such a phenomenon is not restricted to split graphs like P 4 or to graphs with a nontrivial canonical decomposition; for example, in each of the nine labeled realizations of d = (4, 4, 3, 3, 3, 1), the vertices of degree 4 are adjacent, though the Erdős-Gallai difference list ∆(d) = (1, 1, 2, 2) contains no 0 term. The paper [2] shows that the occurrence of a forcible adjacency relationship requires that 0 ≤ ∆ k (d) ≤ 1 for some k; when some minor technicalities are satisfied, the converse is true as well.…”

Weakly threshold graphs

“…Stated another way, in a threshold graph the presence or absence of an edge between two vertices is uniquely determined by the degrees of those two vertices. In a recent paper [1], the author characterized the circumstances under which an edge (or non-edge) is forced to appear in all realizations of a degree sequence. The answer can be stated in terms of the quantities…”

“…for 1 ≤ k ≤ m(d), which we call the Erdős-Gallai differences of d. By Theorem 1.2 the Erdős-Gallai differences are all nonnegative for any degree sequence. As shown in [1], in order for an adjacency relationship to be constant among all labeled realizations of a degree sequence, it is necessary that an Erdős-Gallai difference be at most 1.…”

“…It is perhaps natural to wonder, though, what properties of threshold graphs may continue to hold in a more general form if this condition is relaxed somewhat. In light of the significance of Erdős-Gallai differences of 1, at least in the degree sequence problem of [1], we make a definition.…”