Search CORE

8 research outputs found

The

k

-core of a graph is the largest subgraph of minimum degree at least

k

. We show that for

k

sufficiently large, the

(k + 2)

-core of a random graph \G(n,p) asymptotically almost surely has a spanning

k

-regular subgraph. Thus the threshold for the appearance of a

k

-regular subgraph of a random graph is at most the threshold for the

(k+2)

-core. In particular, this pins down the point of appearance of a

k

-regular subgraph in \G(n,p) to a window for

p

of width roughly

2/n

for large

n

and moderately large

k

We prove that

G_{n,p=c/n}

whp has a

k

-regular subgraph if

c

is at least

e^{-\Theta(k)}

above the threshold for the appearance of a subgraph with minimum degree at least

k

; i.e. an non-empty

k

-core. In particular, this pins down the threshold for the appearance of a

k

-regular subgraph to a window of size

e^{-\Theta(k)}