<s>
In	O
the	O
mathematical	O
field	O
of	O
graph	O
theory	O
,	O
LCF	B-Language
notation	I-Language
or	O
LCF	B-Language
code	I-Language
is	O
a	O
notation	O
devised	O
by	O
Joshua	O
Lederberg	O
,	O
and	O
extended	O
by	O
H	O
.	O
S	O
.	O
M	O
.	O
Coxeter	O
and	O
Robert	O
Frucht	O
,	O
for	O
the	O
representation	O
of	O
cubic	O
graphs	O
that	O
contain	O
a	O
Hamiltonian	O
cycle	O
.	O
</s>
<s>
The	O
cycle	O
itself	O
includes	O
two	O
out	O
of	O
the	O
three	O
adjacencies	O
for	O
each	O
vertex	O
,	O
and	O
the	O
LCF	B-Language
notation	I-Language
specifies	O
how	O
far	O
along	O
the	O
cycle	O
each	O
vertex	O
's	O
third	O
neighbor	O
is	O
.	O
</s>
<s>
A	O
single	O
graph	O
may	O
have	O
multiple	O
different	O
representations	O
in	O
LCF	B-Language
notation	I-Language
.	O
</s>
<s>
The	O
basic	O
form	O
of	O
the	O
LCF	B-Language
notation	I-Language
is	O
just	O
the	O
sequence	O
of	O
these	O
numbers	O
of	O
positions	O
,	O
starting	O
from	O
an	O
arbitrarily	O
chosen	O
vertex	O
and	O
written	O
in	O
square	O
brackets	O
.	O
</s>
<s>
Entries	O
congruent	O
modulo	O
N	O
to	O
0	O
,	O
1	O
,	O
or	O
N1	O
do	O
not	O
appear	O
in	O
this	O
sequence	O
of	O
numbers	O
,	O
because	O
they	O
would	O
correspond	O
either	O
to	O
a	O
loop	O
or	O
multiple	B-Language
adjacency	I-Language
,	O
neither	O
of	O
which	O
are	O
permitted	O
in	O
simple	O
graphs	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
Nauru	O
graph	O
,	O
shown	O
on	O
the	O
right	O
,	O
has	O
four	O
repetitions	O
of	O
the	O
same	O
six	O
offsets	O
,	O
and	O
can	O
be	O
represented	O
by	O
the	O
LCF	B-Language
notation	I-Language
[5,9,7,7,9,5]4	O
.	O
</s>
<s>
A	O
single	O
graph	O
may	O
have	O
multiple	O
different	O
LCF	B-Language
notations	I-Language
,	O
depending	O
on	O
the	O
choices	O
of	O
Hamiltonian	O
cycle	O
and	O
starting	O
vertex	O
.	O
</s>
<s>
LCF	B-Language
notation	I-Language
is	O
useful	O
in	O
publishing	O
concise	O
descriptions	O
of	O
Hamiltonian	O
cubic	O
graphs	O
,	O
such	O
as	O
the	O
examples	O
below	O
.	O
</s>
<s>
In	O
addition	O
,	O
some	O
software	O
packages	O
for	O
manipulating	O
graphs	O
include	O
utilities	O
for	O
creating	O
a	O
graph	O
from	O
its	O
LCF	B-Language
notation	I-Language
.	O
</s>
<s>
If	O
a	O
graph	O
is	O
represented	O
by	O
LCF	B-Language
notation	I-Language
,	O
it	O
is	O
straightforward	O
to	O
test	O
whether	O
the	O
graph	O
is	O
bipartite	O
:	O
this	O
is	O
true	O
if	O
and	O
only	O
if	O
all	O
of	O
the	O
offsets	O
in	O
the	O
LCF	B-Language
notation	I-Language
are	O
odd	O
.	O
</s>
<s>
A	O
more	O
complex	O
extended	O
version	O
of	O
LCF	B-Language
notation	I-Language
was	O
provided	O
by	O
Coxeter	O
,	O
Frucht	O
,	O
and	O
Powers	O
in	O
later	O
work	O
.	O
</s>
