<s>
In	O
graph	O
theory	O
,	O
a	O
weak	B-Operating_System
coloring	I-Operating_System
is	O
a	O
special	O
case	O
of	O
a	O
graph	O
labeling	O
.	O
</s>
<s>
A	O
graph	O
vertex	O
coloring	O
is	O
a	O
weak	B-Operating_System
coloring	I-Operating_System
,	O
but	O
not	O
necessarily	O
vice	O
versa	O
.	O
</s>
<s>
Part	O
(	O
b	O
)	O
shows	O
a	O
breadth-first	B-Algorithm
search	I-Algorithm
tree	O
of	O
the	O
same	O
graph	O
.	O
</s>
<s>
Historically	O
,	O
weak	B-Operating_System
coloring	I-Operating_System
served	O
as	O
the	O
first	O
non-trivial	O
example	O
of	O
a	O
graph	O
problem	O
that	O
can	O
be	O
solved	O
with	O
a	O
local	O
algorithm	O
(	O
a	O
distributed	B-Operating_System
algorithm	I-Operating_System
that	O
runs	O
in	O
a	O
constant	O
number	O
of	O
synchronous	O
communication	O
rounds	O
)	O
.	O
</s>
<s>
More	O
precisely	O
,	O
if	O
the	O
degree	O
of	O
each	O
node	O
is	O
odd	O
and	O
bounded	O
by	O
a	O
constant	O
,	O
then	O
there	O
is	O
a	O
constant-time	O
distributed	B-Operating_System
algorithm	I-Operating_System
for	O
weak	O
2-coloring	O
.	O
</s>
<s>
This	O
is	O
different	O
from	O
(	O
non-weak	O
)	O
vertex	O
coloring	O
:	O
there	O
is	O
no	O
constant-time	O
distributed	B-Operating_System
algorithm	I-Operating_System
for	O
vertex	O
coloring	O
;	O
the	O
best	O
possible	O
algorithms	O
(	O
for	O
finding	O
a	O
minimal	O
but	O
not	O
necessarily	O
minimum	O
coloring	O
)	O
require	O
communication	O
rounds	O
.	O
</s>
