15 research outputs found
A note on the singularity probability of random directed -regular graphs
In this note we show that the singular probability of the adjacency matrix of
a random -regular graph on vertices, where is fixed and , is bounded by . This improves a recent bound by Huang.
Our method is based on the study of the singularity problem modulo a prime
together with an inverse-type result on the decay of the characteristic
function. The latter is related to the inverse Kneser's problem in
combinatorics.Comment: 24 pages, 2 figure
Cycle factors and renewal theory
For which values of does a uniformly chosen -regular graph on
vertices typically contain vertex-disjoint -cycles (a -cycle
factor)? To date, this has been answered for and for ; the
former, the Hamiltonicity problem, was finally answered in the affirmative by
Robinson and Wormald in 1992, while the answer in the latter case is negative
since with high probability most vertices do not lie on -cycles.
Here we settle the problem completely: the threshold for a -cycle factor
in as above is with . Precisely, we prove a 2-point concentration result: if divides then contains a -cycle factor
w.h.p., whereas if then w.h.p. it
does not. As a byproduct, we confirm the "Comb Conjecture," an old problem
concerning the embedding of certain spanning trees in the random graph
.
The proof follows the small subgraph conditioning framework, but the
associated second moment analysis here is far more delicate than in any earlier
use of this method and involves several novel features, among them a sharp
estimate for tail probabilities in renewal processes without replacement which
may be of independent interest.Comment: 45 page