<s>
This	O
was	O
generalized	O
to	O
coloring	O
the	O
faces	O
of	O
a	O
graph	O
embedded	B-Algorithm
in	O
the	O
plane	O
.	O
</s>
<s>
One	O
of	O
the	O
major	O
applications	O
of	O
graph	O
coloring	O
,	O
register	O
allocation	O
in	O
compilers	B-Language
,	O
was	O
introduced	O
in	O
1981	O
.	O
</s>
<s>
If	O
we	O
interpret	O
a	O
coloring	O
of	O
a	O
graph	O
on	O
vertices	O
as	O
a	O
vector	O
in	O
,	O
the	O
action	O
of	O
an	O
automorphism	O
is	O
a	O
permutation	B-Algorithm
of	O
the	O
coefficients	O
in	O
the	O
coloring	O
vector	O
.	O
</s>
<s>
For	O
planar	O
graphs	O
,	O
vertex	O
colorings	O
are	O
essentially	O
dual	O
to	O
nowhere-zero	B-Algorithm
flows	I-Algorithm
.	O
</s>
<s>
Determining	O
if	O
a	O
graph	O
can	O
be	O
colored	O
with	O
2	O
colors	O
is	O
equivalent	O
to	O
determining	O
whether	O
or	O
not	O
the	O
graph	O
is	O
bipartite	O
,	O
and	O
thus	O
computable	O
in	O
linear	O
time	O
using	O
breadth-first	B-Algorithm
search	I-Algorithm
or	O
depth-first	B-Algorithm
search	I-Algorithm
.	O
</s>
<s>
If	O
the	O
graph	O
is	O
planar	O
and	O
has	O
low	O
branch-width	O
(	O
or	O
is	O
nonplanar	O
but	O
with	O
a	O
known	O
branch	O
decomposition	O
)	O
,	O
then	O
it	O
can	O
be	O
solved	O
in	O
polynomial	O
time	O
using	O
dynamic	B-Algorithm
programming	I-Algorithm
.	O
</s>
<s>
Brute-force	B-Algorithm
search	I-Algorithm
for	O
a	O
k-coloring	O
considers	O
each	O
of	O
the	O
assignments	O
of	O
k	O
colors	O
to	O
n	O
vertices	O
and	O
checks	O
for	O
each	O
if	O
it	O
is	O
legal	O
.	O
</s>
<s>
Using	O
dynamic	B-Algorithm
programming	I-Algorithm
and	O
a	O
bound	O
on	O
the	O
number	O
of	O
maximal	O
independent	O
sets	O
,	O
k-colorability	O
can	O
be	O
decided	O
in	O
time	O
and	O
space	O
.	O
</s>
<s>
The	O
running	O
time	O
satisfies	O
the	O
same	O
recurrence	O
relation	O
as	O
the	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
,	O
so	O
in	O
the	O
worst	O
case	O
the	O
algorithm	O
runs	O
in	O
time	O
within	O
a	O
polynomial	O
factor	O
of	O
for	O
n	O
vertices	O
and	O
m	O
edges	O
.	O
</s>
<s>
In	O
practice	O
,	O
branch	B-Algorithm
and	I-Algorithm
bound	I-Algorithm
strategies	O
and	O
graph	O
isomorphism	O
rejection	O
are	O
employed	O
to	O
avoid	O
some	O
recursive	O
calls	O
.	O
</s>
<s>
The	O
greedy	B-Algorithm
algorithm	I-Algorithm
considers	O
the	O
vertices	O
in	O
a	O
specific	O
order	O
,…,	O
and	O
assigns	O
to	O
the	O
smallest	O
available	O
color	O
not	O
used	O
by	O
'	O
s	O
neighbours	O
among	O
,…,,	O
adding	O
a	O
fresh	O
color	O
if	O
needed	O
.	O
</s>
<s>
The	O
maximum	O
(	O
worst	O
)	O
number	O
of	O
colors	O
that	O
can	O
be	O
obtained	O
by	O
the	O
greedy	B-Algorithm
algorithm	I-Algorithm
,	O
by	O
using	O
a	O
vertex	O
ordering	O
chosen	O
to	O
maximize	O
this	O
number	O
,	O
is	O
called	O
the	O
Grundy	O
number	O
of	O
a	O
graph	O
.	O
</s>
<s>
DSatur	O
and	O
RLF	O
are	O
exact	B-Algorithm
for	O
bipartite	O
,	O
cycle	O
,	O
and	O
wheel	O
graphs	O
.	O
</s>
<s>
In	O
the	O
field	O
of	O
distributed	B-Operating_System
algorithms	I-Operating_System
,	O
graph	O
coloring	O
is	O
closely	O
related	O
to	O
the	O
problem	O
of	O
symmetry	O
breaking	O
.	O
</s>
<s>
The	O
current	O
state-of-the-art	O
randomized	O
algorithms	O
are	O
faster	O
for	O
sufficiently	O
large	O
maximum	O
degree	O
Δ	O
than	O
deterministic	B-General_Concept
algorithms	I-General_Concept
.	O
</s>
<s>
In	O
a	O
symmetric	O
graph	O
,	O
a	O
deterministic	B-General_Concept
distributed	B-Operating_System
algorithm	I-Operating_System
cannot	O
find	O
a	O
proper	O
vertex	O
coloring	O
.	O
</s>
<s>
A	O
straightforward	O
distributed	O
version	O
of	O
the	O
greedy	B-Algorithm
algorithm	I-Algorithm
for	O
( Δ+1	O
)	O
-coloring	O
requires	O
Θ(n )	O
communication	O
rounds	O
in	O
the	O
worst	O
case	O
−	O
information	O
may	O
need	O
to	O
be	O
propagated	O
from	O
one	O
side	O
of	O
the	O
network	O
to	O
another	O
side	O
.	O
</s>
<s>
Richard	O
Cole	O
and	O
Uzi	O
Vishkin	O
show	O
that	O
there	O
is	O
a	O
distributed	B-Operating_System
algorithm	I-Operating_System
that	O
reduces	O
the	O
number	O
of	O
colors	O
from	O
n	O
to	O
O(logn )	O
in	O
one	O
synchronous	O
communication	O
step	O
.	O
</s>
<s>
Hence	O
the	O
result	O
by	O
Cole	O
and	O
Vishkin	O
raised	O
the	O
question	O
of	O
whether	O
there	O
is	O
a	O
constant-time	O
distributed	B-Operating_System
algorithm	I-Operating_System
for	O
3-coloring	O
an	O
n-cycle	O
.	O
</s>
<s>
showed	O
that	O
this	O
is	O
not	O
possible	O
:	O
any	O
deterministic	B-General_Concept
distributed	B-Operating_System
algorithm	I-Operating_System
requires	O
Ω(n )	O
communication	O
steps	O
to	O
reduce	O
an	O
n-coloring	O
to	O
a	O
3-coloring	O
in	O
an	O
n-cycle	O
.	O
</s>
<s>
The	O
fastest	O
deterministic	B-General_Concept
algorithms	I-General_Concept
for	O
( Δ+1	O
)	O
-coloring	O
for	O
small	O
Δ	O
are	O
due	O
to	O
Leonid	O
Barenboim	O
,	O
Michael	O
Elkin	O
and	O
Fabian	O
Kuhn	O
.	O
</s>
<s>
Decentralized	O
algorithms	O
are	O
ones	O
where	O
no	O
message	B-Architecture
passing	I-Architecture
is	O
allowed	O
(	O
in	O
contrast	O
to	O
distributed	B-Operating_System
algorithms	I-Operating_System
where	O
local	O
message	B-Architecture
passing	I-Architecture
takes	O
places	O
)	O
,	O
and	O
efficient	O
decentralized	O
algorithms	O
exist	O
that	O
will	O
color	O
a	O
graph	O
if	O
a	O
proper	O
coloring	O
exists	O
.	O
</s>
<s>
The	O
best	O
known	O
approximation	B-Algorithm
algorithm	I-Algorithm
computes	O
a	O
coloring	O
of	O
size	O
at	O
most	O
within	O
a	O
factor	O
O(n(loglogn )	O
2(logn )	O
−3	O
)	O
of	O
the	O
chromatic	O
number	O
.	O
</s>
<s>
There	O
is	O
no	O
FPRAS	B-Algorithm
for	O
evaluating	O
the	O
chromatic	O
polynomial	O
at	O
any	O
rational	O
point	O
k≥	O
1.5	O
except	O
for	O
k	O
=	O
2	O
unless	O
NP	O
=	O
RP	O
.	O
</s>
<s>
In	O
terms	O
of	O
approximation	B-Algorithm
algorithms	I-Algorithm
,	O
Vizing	O
's	O
algorithm	O
shows	O
that	O
the	O
edge	O
chromatic	O
number	O
can	O
be	O
approximated	O
to	O
within	O
4/3	O
,	O
</s>
<s>
These	O
are	O
among	O
the	O
oldest	O
results	O
in	O
the	O
literature	O
of	O
approximation	B-Algorithm
algorithms	I-Algorithm
,	O
even	O
though	O
neither	O
paper	O
makes	O
explicit	O
use	O
of	O
that	O
notion	O
.	O
</s>
<s>
In	O
bandwidth	B-Algorithm
allocation	I-Algorithm
to	O
radio	O
stations	O
,	O
the	O
resulting	O
conflict	O
graph	O
is	O
a	O
unit	O
disk	O
graph	O
,	O
so	O
the	O
coloring	O
problem	O
is	O
3-approximable	O
.	O
</s>
<s>
A	O
compiler	B-Language
is	O
a	O
computer	B-Application
program	I-Application
that	O
translates	O
one	O
computer	B-Language
language	I-Language
into	O
another	O
.	O
</s>
<s>
To	O
improve	O
the	O
execution	O
time	O
of	O
the	O
resulting	O
code	O
,	O
one	O
of	O
the	O
techniques	O
of	O
compiler	B-Application
optimization	I-Application
is	O
register	O
allocation	O
,	O
where	O
the	O
most	O
frequently	O
used	O
values	O
of	O
the	O
compiled	B-Language
program	O
are	O
kept	O
in	O
the	O
fast	O
processor	B-General_Concept
registers	I-General_Concept
.	O
</s>
<s>
The	O
compiler	B-Language
constructs	O
an	O
interference	O
graph	O
,	O
where	O
vertices	O
are	O
variables	O
and	O
an	O
edge	O
connects	O
two	O
vertices	O
if	O
they	O
are	O
needed	O
at	O
the	O
same	O
time	O
.	O
</s>
<s>
The	O
problem	O
of	O
coloring	O
a	O
graph	O
arises	O
in	O
many	O
practical	O
areas	O
such	O
as	O
pattern	B-Language
matching	I-Language
,	O
sports	O
scheduling	O
,	O
designing	O
seating	O
plans	O
,	O
exam	O
timetabling	O
,	O
the	O
scheduling	O
of	O
taxis	O
,	O
and	O
solving	O
Sudoku	O
puzzles	O
.	O
</s>
