<s>
The	O
degeneracy	B-Algorithm
of	O
a	O
graph	O
is	O
the	O
smallest	O
value	O
of	O
k	O
for	O
which	O
it	O
is	O
k-degenerate	O
.	O
</s>
<s>
The	O
degeneracy	B-Algorithm
of	O
a	O
graph	O
is	O
a	O
measure	O
of	O
how	O
sparse	O
it	O
is	O
,	O
and	O
is	O
within	O
a	O
constant	O
factor	O
of	O
other	O
sparsity	O
measures	O
such	O
as	O
the	O
arboricity	O
of	O
a	O
graph	O
.	O
</s>
<s>
Degeneracy	B-Algorithm
is	O
also	O
known	O
as	O
the	O
k-core	B-Algorithm
number	O
,	O
width	O
,	O
and	O
linkage	O
,	O
and	O
is	O
essentially	O
the	O
same	O
as	O
the	O
coloring	B-Algorithm
number	I-Algorithm
or	O
Szekeres	B-Algorithm
–	I-Algorithm
Wilf	I-Algorithm
number	I-Algorithm
(	O
named	O
after	O
)	O
.	O
</s>
<s>
The	O
degeneracy	B-Algorithm
of	O
a	O
graph	O
may	O
be	O
computed	O
in	O
linear	O
time	O
by	O
an	O
algorithm	O
that	O
repeatedly	O
removes	O
minimum-degree	O
vertices	O
.	O
</s>
<s>
The	O
connected	O
components	O
that	O
are	O
left	O
after	O
all	O
vertices	O
of	O
degree	O
less	O
than	O
k	O
have	O
been	O
(	O
repeatedly	O
)	O
removed	O
are	O
called	O
the	O
k-cores	B-Algorithm
of	O
the	O
graph	O
and	O
the	O
degeneracy	B-Algorithm
of	O
a	O
graph	O
is	O
the	O
largest	O
value	O
k	O
such	O
that	O
it	O
has	O
a	O
k-core	B-Algorithm
.	O
</s>
<s>
Every	O
finite	O
planar	O
graph	O
has	O
a	O
vertex	O
of	O
degree	O
five	O
or	O
less	O
;	O
therefore	O
,	O
every	O
planar	O
graph	O
is	O
5-degenerate	O
,	O
and	O
the	O
degeneracy	B-Algorithm
of	O
any	O
planar	O
graph	O
is	O
at	O
most	O
five	O
.	O
</s>
<s>
Similarly	O
,	O
every	O
outerplanar	O
graph	O
has	O
degeneracy	B-Algorithm
at	O
most	O
two	O
,	O
and	O
the	O
Apollonian	O
networks	O
have	O
degeneracy	B-Algorithm
three	O
.	O
</s>
<s>
Every	O
k-regular	O
graph	O
has	O
degeneracy	B-Algorithm
exactlyk	O
.	O
</s>
<s>
More	O
strongly	O
,	O
the	O
degeneracy	B-Algorithm
of	O
a	O
graph	O
equals	O
its	O
maximum	O
vertex	O
degree	O
if	O
and	O
only	O
if	O
at	O
least	O
one	O
of	O
the	O
connected	O
components	O
of	O
the	O
graph	O
is	O
regular	O
of	O
maximum	O
degree	O
.	O
</s>
<s>
For	O
all	O
other	O
graphs	O
,	O
the	O
degeneracy	B-Algorithm
is	O
strictly	O
less	O
than	O
the	O
maximum	O
degree	O
.	O
</s>
<s>
The	O
coloring	B-Algorithm
number	I-Algorithm
of	O
a	O
graph	O
G	O
was	O
defined	O
by	O
to	O
be	O
the	O
least	O
for	O
which	O
there	O
exists	O
an	O
ordering	O
of	O
the	O
vertices	O
of	O
G	O
in	O
which	O
each	O
vertex	O
has	O
fewer	O
than	O
neighbors	O
that	O
are	O
earlier	O
in	O
the	O
ordering	O
.	O
</s>
<s>
It	O
should	O
be	O
distinguished	O
from	O
the	O
chromatic	O
number	O
of	O
G	O
,	O
the	O
minimum	O
number	O
of	O
colors	O
needed	O
to	O
color	O
the	O
vertices	O
so	O
that	O
no	O
two	O
adjacent	O
vertices	O
have	O
the	O
same	O
color	O
;	O
the	O
ordering	O
which	O
determines	O
the	O
coloring	B-Algorithm
number	I-Algorithm
provides	O
an	O
order	O
to	O
color	O
the	O
vertices	O
of	O
G	O
with	O
the	O
coloring	B-Algorithm
number	I-Algorithm
,	O
but	O
in	O
general	O
the	O
chromatic	O
number	O
may	O
be	O
smaller	O
.	O
</s>
<s>
The	O
degeneracy	B-Algorithm
of	O
a	O
graph	O
G	O
was	O
defined	O
by	O
as	O
the	O
least	O
k	O
such	O
that	O
every	O
induced	O
subgraph	O
of	O
G	O
contains	O
a	O
vertex	O
with	O
k	O
or	O
fewer	O
neighbors	O
.	O
</s>
<s>
The	O
two	O
concepts	O
of	O
coloring	B-Algorithm
number	I-Algorithm
and	O
degeneracy	B-Algorithm
are	O
equivalent	O
:	O
in	O
any	O
finite	O
graph	O
the	O
degeneracy	B-Algorithm
is	O
just	O
one	O
less	O
than	O
the	O
coloring	B-Algorithm
number	I-Algorithm
.	O
</s>
<s>
For	O
,	O
if	O
a	O
graph	O
has	O
an	O
ordering	O
with	O
coloring	B-Algorithm
number	I-Algorithm
κ	O
then	O
in	O
each	O
subgraph	O
H	O
the	O
vertex	O
that	O
belongs	O
to	O
H	O
and	O
is	O
last	O
in	O
the	O
ordering	O
has	O
at	O
most	O
1	O
neighbors	O
in	O
H	O
.	O
In	O
the	O
other	O
direction	O
,	O
if	O
G	O
is	O
k-degenerate	O
,	O
then	O
an	O
ordering	O
with	O
coloring	B-Algorithm
number	I-Algorithm
k+1	O
can	O
be	O
obtained	O
by	O
repeatedly	O
finding	O
a	O
vertex	O
v	O
with	O
at	O
most	O
k	O
neighbors	O
,	O
removing	O
v	O
from	O
the	O
graph	O
,	O
ordering	O
the	O
remaining	O
vertices	O
,	O
and	O
adding	O
v	O
to	O
the	O
end	O
of	O
the	O
order	O
.	O
</s>
<s>
A	O
third	O
,	O
equivalent	O
formulation	O
is	O
that	O
G	O
is	O
k-degenerate	O
(	O
or	O
has	O
coloring	B-Algorithm
number	I-Algorithm
at	O
most	O
k+1	O
)	O
if	O
and	O
only	O
if	O
the	O
edges	O
of	O
G	O
can	O
be	O
oriented	O
to	O
form	O
a	O
directed	O
acyclic	O
graph	O
with	O
outdegree	O
at	O
most	O
k	O
.	O
Such	O
an	O
orientation	O
can	O
be	O
formed	O
by	O
orienting	O
each	O
edge	O
towards	O
the	O
earlier	O
of	O
its	O
two	O
endpoints	O
in	O
a	O
coloring	B-Algorithm
number	I-Algorithm
ordering	O
.	O
</s>
<s>
In	O
the	O
other	O
direction	O
,	O
if	O
an	O
orientation	O
with	O
outdegree	O
k	O
is	O
given	O
,	O
an	O
ordering	O
with	O
coloring	B-Algorithm
number	I-Algorithm
k+1	O
can	O
be	O
obtained	O
as	O
a	O
topological	B-Algorithm
ordering	I-Algorithm
of	O
the	O
resulting	O
directed	O
acyclic	O
graph	O
.	O
</s>
<s>
A	O
k-core	B-Algorithm
of	O
a	O
graph	O
G	O
is	O
a	O
maximal	O
connected	O
subgraph	O
of	O
G	O
in	O
which	O
all	O
vertices	O
have	O
degree	O
at	O
least	O
k	O
.	O
Equivalently	O
,	O
it	O
is	O
one	O
of	O
the	O
connected	O
components	O
of	O
the	O
subgraph	O
of	O
G	O
formed	O
by	O
repeatedly	O
deleting	O
all	O
vertices	O
of	O
degree	O
less	O
than	O
k	O
.	O
If	O
a	O
non-empty	O
k-core	B-Algorithm
exists	O
,	O
then	O
,	O
clearly	O
,	O
G	O
has	O
degeneracy	B-Algorithm
at	O
least	O
k	O
,	O
and	O
the	O
degeneracy	B-Algorithm
of	O
G	O
is	O
the	O
largest	O
k	O
for	O
which	O
G	O
has	O
a	O
k-core	B-Algorithm
.	O
</s>
<s>
The	O
concept	O
of	O
a	O
k-core	B-Algorithm
was	O
introduced	O
to	O
study	O
the	O
clustering	O
structure	O
of	O
social	O
networks	O
and	O
to	O
describe	O
the	O
evolution	O
of	O
random	O
graphs	O
.	O
</s>
<s>
Bootstrap	O
percolation	O
is	O
a	O
random	O
process	O
studied	O
as	O
an	O
epidemic	O
model	O
and	O
as	O
a	O
model	O
for	O
fault	B-General_Concept
tolerance	I-General_Concept
for	O
distributed	B-Architecture
computing	I-Architecture
.	O
</s>
<s>
outline	O
an	O
algorithm	O
to	O
derive	O
the	O
degeneracy	B-Algorithm
ordering	O
of	O
a	O
graph	O
with	O
vertex	O
set	O
and	O
edge	O
set	O
in	O
time	O
and	O
space	O
,	O
by	O
storing	O
vertices	O
in	O
a	O
degree-indexed	O
bucket	B-Application
queue	I-Application
and	O
repeatedly	O
removing	O
the	O
vertex	O
with	O
the	O
smallest	O
degree	O
.	O
</s>
<s>
The	O
degeneracy	B-Algorithm
is	O
given	O
by	O
the	O
highest	O
degree	O
of	O
any	O
vertex	O
at	O
the	O
time	O
of	O
its	O
removal	O
.	O
</s>
<s>
Thus	O
,	O
the	O
arboricity	O
of	O
G	O
is	O
at	O
most	O
equal	O
to	O
its	O
degeneracy	B-Algorithm
.	O
</s>
<s>
In	O
the	O
other	O
direction	O
,	O
an	O
n-vertex	O
graph	O
that	O
can	O
be	O
partitioned	O
into	O
k	O
forests	O
has	O
at	O
most	O
k(n1 )	O
edges	O
and	O
therefore	O
has	O
a	O
vertex	O
of	O
degree	O
at	O
most	O
2k1	O
–	O
thus	O
,	O
the	O
degeneracy	B-Algorithm
is	O
less	O
than	O
twice	O
the	O
arboricity	O
.	O
</s>
<s>
The	O
edges	O
of	O
a	O
graph	O
with	O
such	O
an	O
orientation	O
may	O
be	O
partitioned	O
in	O
the	O
same	O
way	O
into	O
k	O
pseudoforests	O
,	O
and	O
conversely	O
any	O
partition	O
of	O
a	O
graph	O
's	O
edges	O
into	O
k	O
pseudoforests	O
leads	O
to	O
an	O
outdegree-k	O
orientation	O
(	O
by	O
choosing	O
an	O
outdegree-1	O
orientation	O
for	O
each	O
pseudoforest	O
)	O
,	O
so	O
the	O
minimum	O
outdegree	O
of	O
such	O
an	O
orientation	O
is	O
the	O
pseudoarboricity	O
,	O
which	O
again	O
is	O
at	O
most	O
equal	O
to	O
the	O
degeneracy	B-Algorithm
.	O
</s>
<s>
The	O
thickness	O
is	O
also	O
within	O
a	O
constant	O
factor	O
of	O
the	O
arboricity	O
,	O
and	O
therefore	O
also	O
of	O
the	O
degeneracy	B-Algorithm
.	O
</s>
<s>
the	O
maximum	O
clique	O
,	O
the	O
latter	O
invariant	O
is	O
also	O
at	O
most	O
degeneracy	B-Algorithm
plus	O
one	O
.	O
</s>
<s>
By	O
using	O
a	O
greedy	O
coloring	O
algorithm	O
on	O
an	O
ordering	O
with	O
optimal	O
coloring	B-Algorithm
number	I-Algorithm
,	O
one	O
can	O
graph	O
color	O
a	O
k-degenerate	O
graph	O
using	O
at	O
most	O
k+1	O
colors	O
.	O
</s>
<s>
Since	O
these	O
paths	O
must	O
leave	O
the	O
two	O
vertices	O
of	O
the	O
pair	O
via	O
disjoint	O
edges	O
,	O
a	O
k-vertex-connected	O
graph	O
must	O
have	O
degeneracy	B-Algorithm
at	O
least	O
k	O
.	O
Concepts	O
related	O
to	O
k-cores	B-Algorithm
but	O
based	O
on	O
vertex	O
connectivity	O
have	O
been	O
studied	O
in	O
social	O
network	O
theory	O
under	O
the	O
name	O
of	O
structural	O
cohesion	O
.	O
</s>
<s>
Therefore	O
,	O
the	O
degeneracy	B-Algorithm
is	O
at	O
most	O
equal	O
to	O
the	O
treewidth	O
and	O
at	O
most	O
equal	O
to	O
the	O
pathwidth	O
.	O
</s>
<s>
However	O
,	O
there	O
exist	O
graphs	O
with	O
bounded	O
degeneracy	B-Algorithm
and	O
unbounded	O
treewidth	O
,	O
such	O
as	O
the	O
grid	O
graphs	O
.	O
</s>
<s>
The	O
Burr	O
–	O
Erdős	O
conjecture	O
relates	O
the	O
degeneracy	B-Algorithm
of	O
a	O
graph	O
G	O
to	O
the	O
Ramsey	O
number	O
of	O
G	O
,	O
the	O
least	O
n	O
such	O
that	O
any	O
two-edge-coloring	O
of	O
an	O
n-vertex	O
complete	O
graph	O
must	O
contain	O
a	O
monochromatic	O
copy	O
of	O
G	O
.	O
Specifically	O
,	O
the	O
conjecture	O
is	O
that	O
for	O
any	O
fixed	O
value	O
of	O
k	O
,	O
the	O
Ramsey	O
number	O
of	O
k-degenerate	O
graphs	O
grows	O
linearly	O
in	O
the	O
number	O
of	O
vertices	O
of	O
the	O
graphs	O
.	O
</s>
<s>
Although	O
concepts	O
of	O
degeneracy	B-Algorithm
and	O
coloring	B-Algorithm
number	I-Algorithm
are	O
frequently	O
considered	O
in	O
the	O
context	O
of	O
finite	O
graphs	O
,	O
the	O
original	O
motivation	O
for	O
was	O
the	O
theory	O
of	O
infinite	O
graphs	O
.	O
</s>
<s>
For	O
an	O
infinite	O
graph	O
G	O
,	O
one	O
may	O
define	O
the	O
coloring	B-Algorithm
number	I-Algorithm
analogously	O
to	O
the	O
definition	O
for	O
finite	O
graphs	O
,	O
as	O
the	O
smallest	O
cardinal	O
number	O
such	O
that	O
there	O
exists	O
a	O
well-ordering	O
of	O
the	O
vertices	O
of	O
G	O
in	O
which	O
each	O
vertex	O
has	O
fewer	O
than	O
α	O
neighbors	O
that	O
are	O
earlier	O
in	O
the	O
ordering	O
.	O
</s>
<s>
The	O
degeneracy	B-Algorithm
of	O
random	O
subsets	O
of	O
infinite	O
lattices	O
has	O
been	O
studied	O
under	O
the	O
name	O
of	O
bootstrap	O
percolation	O
.	O
</s>
