Colloquium publications.

16

THE CAMBRIDGE COLLOQUIUM.

by any finite set of points on it into a 1-circuit. Conversely, it
is easy to see that the set of all points on a I-circuit is a simple
closed curve. It is obvious, further, that any linear graph such
that each vertex is an end of two and only two 1-cells is either a
1-circuit or a set of 1-circuits no two of which have a point in
common.
Consider a linear graph C1 such that each vertex is an end of
an even number of edges. Let us denote by 2ni the number of
edges incident with each vertex ai. The edges incident with
each vertex ai~ may be grouped arbitrarily in ni pairs no two of
which have an edge in common; let these pairs of edges be called
the pairs associated with the vertex ai. Let C1' be a graph
coincident with C1 in such a way that (1) there is one and only
one point of C1' on each point of C1 which is not a vertex and
(2) there are ni vertices of C1' on each vertex ao of C1 each of
these vertices of C1' being incident only with the two edges of C1'
which coincide with a pair associated with a?.
The linear graph C1' has just two edges incident with each of
its vertices and therefore consists of a number of 1-circuits.
Each of these 1-circuits is coincident with a 1-circuit of Ci, and
no two of the 1-circuits of C1 thus determined have a 1-cell in
common. Hence C1 consists of a number of 1-circuits which
have only a finite number of O-cells in common.
It is obvious that a linear graph composed of a number of
closed curves having only a finite number of points in common
has an even number of 1-cells incident with each vertex. Hence
a necessary and sufficient condition that C1 consist of a number of
1-circuits having only 0-cells in common is that each 0-cell of C1
be incident with an even number of 1-cells. A set of 1-circuits
having only O-cells in common will be referred to briefly as a set
of 1-circuits.
23. The sum of the symbols (xl, x2, ', xa) for the O-circuits
which bound the 1-cells of a 1-circuit is (0, 0,.., 0) because each
0-cell appears in two and only two of these O-circuits. Hence
any 1-circuit or set of 1-circuits determines a linear relation,
modulo 2, among the bounding O-circuits.