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ABSTRACT In 1956, Tutte proved the celebrated theorem that every 4‐connected planar graph is Hamiltonian. This result implies that every more than ‐tough planar graph on at least three vertices is Hamiltonian and so has a 2‐factor. Owens in 1999 constructed non‐Hamiltonian maximal planar graphs of toughness arbitrarily close to and asked whether there exists a maximal non‐Hamiltonian planar graph of toughness exactly . In fact, the graphs Owens constructed do not even contain a 2‐factor. Thus the toughness of exactly is the only case left in asking the existence of 2‐factors in tough planar graphs. This question was also asked by Bauer, Broersma, and Schmeichel in a survey. In this paper, we close this gap by constructing a maximal ‐tough plane graph with no 2‐factor, answering the question asked by Owens as well as by Bauer, Broersma, and Schmeichel.
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PLANAR MINIMAL SURFACES WITH POLYNOMIAL GROWTH IN THE Sp(4,R)-SYMMETRIC SPACE
We study the asymptotic geometry of a family of conformally planar minimal surfaces with polynomial growth in the Sp(4,R)-symmetric space. We describe a homeomorphism between the “Hitchin component” of wild Sp(4,R)- Higgs bundles over CP1 with a single pole at infinity and a component of maximal surfaces with light-like polygonal boundary in H2,2. Moreover, we identify those surfaces with convex embeddings into the Grassmannian of symplectic planes of R4. We show, in addition, that our planar maximal surfaces are the local limits of equivariant maximal surfaces in H2,2 associated to Sp(4,R)-Hitchin representations along rays of holomorphic quartic differentials.
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- PAR ID:
- 10432511
- Date Published:
- Journal Name:
- American journal of mathematics
- ISSN:
- 1080-6377
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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