VOL. 54, 1965 MATHEMATICS: S. LEFSCHETZ 1763

1 Morse, M., "Relations between the critical points of a real function of n variables," Trans. Amer. Math. Soc., 27, 34, Lemma 4 (1925). 2 Evaluation at the origin is indicated by a superscript 0. 3Morse, M., "Functional topology and abstract variational theory," Memorial des Sciences Math. (Paris: Gauthier-Villars, 1938).

PLANAR GRAPHS AND RELATED TOPICS* BY SOLOMON LEFSCHETZ

DEPARTMENT OF MATHEMATICS, PRINCETON UNIVERSITY, AND CENTER FOR DYNAMICAL SYSTEMS, BROWN UNIVERSITY Communicated October 14, 1965 1. A planar graph G is a graph which may be topologically imbedded in a plane or equivalently in a sphere. N.a.s.c. for this to happen have been given by Kuratowski,1 Hassler Whitney,2 and Saunders Mac Lane.3 The simplest formula- tion is Mac Lane's who has proved a particularly elegant theorem. We propose to give here a new and more rapid proof of this proposition-more rapid because of the utilization of some well-known results from two-dimensional algebraic topology. 2. A graph G, as we understand it, is a finite collection of vertices a1, a2, aa, and disjoint arcs b1, b2, .. ., ba, which is connected and such that every arc joins two distinct vertices. The characteristic x(G) of G satisfies

x(G) = ao - al = 1-R. Here R is the first Betti number of G. By a loop X of G is meant an oriented Jordan curve made up of closed arcs of G. The loop X thus determines a unique one-cycle also designated for convenience by X. We prove with little difficulty: there is a base for the one-cycles made up of loops X1, * * , XR. 3. If K is a cell-complex of any dimension, let IKI denote the polyhedron which is the set of all points in all cells of K. Suppose that K is a two-complex with ah, h-cells, h = 0, 1, 2. Its characteristic

X(K) = ao - a, + a2 depends solely on IKI and so may as well be written X(JKI).4 We say that K covers IKI. We shall be concerned almost exclusively with complexes covering a surface (two-manifold). Let then K cover a surface and let ch be its oriented two-cells and Xh = 5ch, the chain-boundary loop of Ch. Let also r be the graph made up of the closed arcs of K. One recognizes at once the following properties of K phrased in terms of r: (A) F is connected5 (hence a graph) and nonseparable (no removal of a vertex disconnects r). (B) Every arc of r belongs to exactly two loops X,. Downloaded by guest on September 29, 2021 1764 MATHEMATICS: S. LEFSCHETZ PROC. N. A. S.

(C) Umbrella property: The arcs bi', b2', ..., and loops Xi', X2', ..., containing any vertex a of r may be ranged in a single circular system or "umbrella" bi'Xl'b2' ... b/X/,'bl' such that bt', bh+1' are arcs of Xh(b,+1' = bit). The next property admits of two possibilities (translation of orientability or nonorientability of K): (D1) Orientable F: The loops Xh may be so oriented that AXh = 0, this being the only (independent) relation which they satisfy. (D2) Nonorientable r: The loops Xh are independent. A graph with properties A, B, Di may possess loops independent of the Xh; it is then natural to introduce for it the following character:

excess E = R(G) - (a2 -1)[= R(G) - a2] if G is orientable [nonorientable]. Another independent character is the deficiency 0, or number of excessive um- brellas, that is, which contradict C. Thus, if the vertex a has 1 + k umbrellas, its contribution to 0 is k. 5. THEOREM OF SAUNDERS MAC LANE. N.a.s.c. in order that a nonseparable graph G with first Betti number R on reals be planar is that it contain a set of loops X0, X1, .. ., XR such that: (a) every arc of G belongs to exactly two loops of the set; (b) the only independent chain-relation between the loops of the set is 2Xh = 0. Assuming G on a sphere S one verifies at once necessity. Suppose now that the conditions are fulfilled. Let KG be the complex obtained by spanning the Xh with two-cells ch.6 Under our assumptions every nonvertex point of 1KG! has for neighborhood a two-cell. An umbrella of G, say of center ah, gives rise in IKG! to a two-cell or disk containing ah. Hence, if ah is the center of 1 + Ph umbrellas, it has in IKG! a neighborhood made up of 1 + Ph disks with one common point. Thus, IKG! is a surface if and only if everyph = 0, that is, if 0 = 2pPh = 0. We replace KG by a complex K whose arcs and two-cells are disposed as in KG but with just one umbrella at each vertex. Thus 1K! will be a surface. The first step in the construction of K is as follows: let ui, ..., up, be the umbrellas centered at a,. Replace a, by new vertices a1,,, h = 1, 2, . . ., pi leaving all but the uh and attached two-cells untouched. Replace now uh by an umbrella uh' and disk centered at alh and with elements disposed like those of uh plus disk. Let K1 be the resulting new complex. We will now show that pi = 0, hence likewise every Ph = 0 = 0, so that KG is a surface. Suppose that pi t 0. Let G1 be the graph made up of the closed arcs of K1. Since G is nonseparable, Gi is connected. Hence, a11 and a12 may be joined in G1 by an arc u. To this arc there corresponds a loop X in G obtained from the coincidence of al2 with all. Since 1A is lost in the passage G - G1, we have R(G1) < R(G) = R. But obviously Xo, X1, . . ., XR are still present and independent in Gi. Hence, R(G1) _ R. This contradiction proves our assertion that KG is a surface. By direct calculation, however, x(KG) = x(G) + 1 + R = 1-1 + (1 + R) = 2. Hence, KG covers a sphere S in which G is imbedded. This completes the proof of the theorem. Note: Saunders i\Lac Lane did not use the sphere and so his formulation differs from ours, property B being replaced by "every arc belongs to one or two loops X,,." 6. The generalization of the preceding result offers no difficulty. This time, Downloaded by guest on September 29, 2021 VOL. 54, 1965 PATHOLOGY: E. WEILER 1765 one may not use the characteristic to show that C is a consequence of A, B, Di and so one obtains more or less by the same reasoning: THEOREM. N.a.s.c. in order that a connected, nonseparable graph G of Betti number R be imbeddable in an orientable surface of characteristic 2 - e (e even) [in a nonorientable surface of characteristic 1 - e] is that it possess 1 + R- e loops Xo, Xi, . . ., XR-E satisfying the only independent chain-relation 2Xh = 0 [R -e inde- pendent loops X1, . . ., XR-] such that every arc of the graph belongs to exactly two loops of the set X0, .. ., XR- [X1N ... XRJ]. Moreover, at each vertex of the graph property C (umbrella uniqueness) holds relative to the respective set of loops. Examples: Imbedding in a projective plane: e = 1, with D2, (x = 1); imbedding in a torus; e = 2 with Dl(x = 0). The author wishes to express his appreciation for valuable remarks received from Hassler Whitney and Saunders Mac Lane. * Supported in part by AFSOR under contract 693-64. 1 Kuratowski, K., "Sur le problbme des courbes gauches en topologie," Fund. Math., 15, 271- 283 (1930). 2Whitney, H., "Nonseparable and planar graphs," Trans. Amer. Math. Soc., 34, 331-362 (1932); Whitney, H., "Planar graphs," Fund. Math., 24, 23-34 (1933). 3 Mac Lane, S., "A combinatorial condition for planar graphs," Fund Math., 28, 22-32 (1936); Mac Lane; S., "A structural characterization of planar combinatorial graphs," Duke Math J., 3, 460-472 (1937). 4 The proof which is not elementary may be found in the author's book Introduction to Topology. 6For IKI is connected as a surface and this readily implies the same for r. 6 Note that the connectedness of G implies that of |KG!.

DIFFERENTIAL ACTIVITY OF ALLELIC -y-GLOBULIN GENES IN ANTIBODY-PRODUCING CELLS* BY EBERHARDT WEILER THE INSTITUTE FOR CANCER RESEARCH, PHILADELPHIA Communicated by Thomas F. Anderson, August 30, 1965 For heterozygous individuals the question may be asked whether at a given locus both alleles are active in individual cells, or only one. A decision can be reached when methods are available to determine phenotypes at the cellular level. This has been done with glucose-6-phosphodehydrogeilase, specified by an X- linked gene, and it has been found that in each cell of heterozygous females only one or the other allele is phenotypically expressed.' 2 Such "phenotypic mosaicism" has not been found with markers on autosomes.' The present work is concerned with an autosomal locus in mice,3 which is active in the synthesis of y-globulin. In mice, gamma-globulin (of the 7S'y2a class)4 occurs in genetically segregating variant forms (allotypes). In a preceding paper, a new effect called facilitation of hemolysis was described, which makes it possible to determine the allotype of anti- bodies secreted by individual cells.5 In experiments to be described in this paper, hemolysis facilitation was applied to study heterozygous cells, and it will be shown that in single cells only one of the two parental phenotypes can be recognized. Downloaded by guest on September 29, 2021