Limits and Regularity of Graph Sequences

Thumbnail Image
Lane, Michael
Vempala, Santosh S.
Associated Organization(s)
Organizational Unit
Organizational Unit
Supplementary to
A limit of a sequence of graphs is an object that encodes approximate combinatorial information of the sequence. [Lovasz et al citation, 2008] defines such a limit for sequences of dense graph. For example, if a dense graph sequence (Gn) converges to a graphon W; then en = e(Gn)/v(Gn)² ϵ [0; 1=2) (the number of edges of Gn relative to its number of nodes) converges to some e; and e is directly computable from W: As a second example, if Mn denotes the size of the maximum cut of Gn; and mn = Mn/v(Gn)2 ϵ [0; 1=2) is this size relative to the number of nodes, then mn converges as n → ∞; and once again this limit is directly computable from W: If (Gn) is any less-than-dense sequence, the limit W is still well-defined, but W = 0; which does not carry any information. To account for this, [Benjamini and Schramm citation] have defined a different kind of limit for sparse graph sequences. However, the limit is non-sensical for any greater-than-sparse sequence. This research attempts to fill the gap, by defining limits for a variety of intermediate degree sequences, those strictly between being sparse and being dense. Given a graph N; does there exist a graph M such that the 1-neighborhoods of every vertex of M are isomorphic to N? When such an M exists, it is called a mosaic of N: Bulitko (1977) proved this question to be algorithmically undecidable. We therefore consider a slightly different question: Given a graph N and assuming it has a mosaic, is that mosaic unique, or if not, what characterizes the collection of all such mosaics? A resolution of this question could have applications to sparse regularity lemmas, analogous to Szemeredi's famous regularity lemma for sparse graphs.
Date Issued
Resource Type
Resource Subtype
Undergraduate Thesis
Rights Statement
Rights URI