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Asymptotic Normality of Some Graph Sequences
Authors:
David Galvin
Institution:
1.Department of Mathematics,University of Notre Dame,Notre Dame,USA
Abstract:
For a simple finite graph
G
denote by
Open image in new window
G into
k
non-empty independent sets (that is, into classes that span no edges of
G
). If
\(E_n\)
is the graph on
n
vertices with no edges then
Open image in new window
Open image in new window
Open image in new window
graph Stirling number. Harper showed that the sequence of Stirling numbers of the second kind, and thus the graph Stirling sequence of
\(E_n\)
, is
asymptotically normal
—essentially, as
n
grows, the histogram of
Open image in new window
\((G_n)_{n \ge 0}\)
of graphs is there asymptotic normality of
Open image in new window
n,
\(G_n\)
is acyclic and has
n
vertices, then asymptotic normality occurs, and they gave a proof under the added condition that
\(G_n\)
has no more than
\(o(\sqrt{n/\log n})\)
components. Here we settle Thanh and Galvin’s conjecture in the affirmative, and significantly extend it, replacing “acyclic” in their conjecture with “co-chromatic with a quasi-threshold graph, and with negligible chromatic number”. Our proof combines old work of Navon and recent work of Engbers, Galvin and Hilyard on the normal order problem in the Weyl algebra, and work of Kahn on the matching polynomial of a graph.
Keywords:
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