It is proved that any edge of a Pconnected non-planar graph G of order a t least 6 lies in a subdivision of K3,3 in G. For any 3-connected non-planar graph G of order a t least 6 we show that G contains at most four edges which belong to no subdivisions of K3,3 in G.
Kuratowski-Pontrjagin theorem on planar graphs
โ Scribed by Michael Burstein
- Publisher
- Elsevier Science
- Year
- 1978
- Tongue
- English
- Weight
- 257 KB
- Volume
- 24
- Category
- Article
- ISSN
- 0095-8956
No coin nor oath required. For personal study only.
๐ SIMILAR VOLUMES
We present a new short combinatorial proof of the sufficiency part of the well-known Kuratowski's graph planarity criterion. The main steps are to prove that for a minor minimal non-planar graph G and any edge xy: (1) G-x-y does not contain ฮธ-subgraph; (2) G-x-y is homeomorphic to the circle; (3)
C. Thomassen extended Tutte's theorem on cycles in planar graphs in the paper "A Theorem on Paths in Planar Graphs". This note corrects a flaw in his proof.
We prove a theorem on paths with prescribed ends in a planar graph which extends Tutte's theorem on cycles in planar graphs [9] and implies the conjecture of Plummer (51 asserting that every 4-connected planar graph is Hamiltonian-connected.
Re&ved 4 Fkbruary