This note gives a short and elementary proof of MacLane's theorem on the embedding of graphs in a 2-sphere

Oneil, P. V.

English

NASA Technical Reports Server

Abstract
This note gives a short and elementary proof of MacLane's
theorem on the embedding of graphs in a 2-sphere.
?(NASA-CR-127463) A SHORT PROOF OF
MACLANE'S PLANARITY THEOREM P.V. ONeil N72-28589
(College of William and Mary) 1970 4 
.CSCL .12A Unclas
___~ ~G3/19 35389
Reproduced by
NATIONAL TECHNICAL
- INFORMATION SERVICE'i-
-:, . _ DeportmentfCrerc : ..o
'. SpringfieldVA322151
-.!
I
https://ntrs.nasa.gov/search.jsp?R=19720020939 2020-03-11T18:22:33+00:00Z
1.
The purpose of this note is to give a short and elementary
proof of a theorem by Saunders MacLane on the embedding of graphs
in the 2-sphere L4] . Existing proofs are the original ones of
MacLane and an algebraic topology proof by Lefschetz L31 . Our
proof is by Kuratowski's theorem [2] .
Terminology follows L41 and [11 , with the exception that we
shall call MacLane's 2-fold complete set of circuits a P-base.
Let G be a nonseparable graph.
Theorem. If G has a P-base, then G is planar.
Proof. Let C1,...,Cn form a P-base for G, and suppose that G
is nonplanar. Then n)> 1 and, by Kuratowski's theorem, G has a subgraph
H homeomorphic to K5 or to K3 ,3. We claim that H also has a P-base.
This is immediate by induction if it is first shown that G - e has a
P-base for each arc e of G. But, if e is in exactly one Ci, say C1,
then C2, *,Cn form a P-base for G - e, and if e is in two Ci's, say
C1 and C2, then C3 ... ,C, C+ Ci form a P-base for G - e.
Thus H, hence also K5 or K33, has a P-base. We now show that5 3,3,
this is impossible.
If C1 ,...,C6 form a P-base for K5, then each of the ten branches
of K5 is in exactly two of the circuits C,,...,C6 , C7 = XC i. But
each circuit has at least three branches, so
AMS 1970 subject classifications. Primary 05 C 10.
Key words and phrases. planar graphs, 2-fold complete set of circuits.
2Z (number of branches in Ci) = 20 21.
Similarly, if C1 ,...,C4 form a P-base for K3,3, then set
C5 = Ci. Since each circuit in K3, 3 has at least four arcs, thenAn,~~~~~~~,
(number of branches in Ci) = 18> 20,
completing the proof.
The converse of the theorem is of course also true, but the
proof of this is trivial.
3.
References
1. F. Harary, Graph Theory, Addison-Wesley, Reading, Mass., 1969.
2. C. Kuratowski, Sur le probleme des courbes gauches en topologie,
Fund. Math., 15, 1930, 271-283.
3. S. Lefschetz, Planar graphs and related topics, Proc. Nat'l.
Acad. Sci., U.S.A., 54, 1965, 1763-1765.
4. S. MacLane, A combinatorial condition for planar graphs, Fund.
Math., 28, 1937, 22-32.


A short proof of MacLane's planarity theorem

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server