• Corpus ID: 96463185

Theory on Structure and Coloring of Maximal Planar Graphs (I): Relationship between Structure and Coloring

@article{Xu2012TheoryOS,
  title={Theory on Structure and Coloring of Maximal Planar Graphs (I): Relationship between Structure and Coloring},
  author={Jin Xu},
  journal={arXiv: General Mathematics},
  year={2012}
}
  • Jin Xu
  • Published 16 October 2012
  • Mathematics
  • arXiv: General Mathematics
Maximal planar graph refers to the planar graph with the most edges, which means no more edges can be added so that the resulting graph is still planar. The Four-Color Conjecture says that every planar graph without loops is 4-colorable. Indeed, in order to prove Four-Color Conjecture, it clearly suffices to show that all maximal planar graphs are 4-colorable. Since this conjecture was proposed in 1852, no mathematical proofs have been invented up until now. Maybe the main reasons lie in the… 

References

SHOWING 1-10 OF 36 REFERENCES

On generating planar graphs

Hamiltonian cycles through prescribed edges of 4-connected maximal planar graphs

Hamiltonian cycles in planar triangulations with no separating triangles

It is shown that the conclusion of Whitney's theorem still holds if the chords satisfy a certain sparse-ness condition and that a Hamiltonian cycle through a graph satisfying this condition can be found in linear time.

On the number of hamiltonian cycles in a maximal planar graph

A p-vertex maximal planar graph containing exactly four Hamiltonian cycles for every p ≥ 12 vertices is constructed and it is proved that every 4-connected maximalPlanar graph on p vertices contains at least p/(log2 p) Hamiltoniancycles.

Tow Maximal Planar Graphs with Path Bichromatic Subgraph Only.

In this paper, it is proved that every bichromatic subgraph of every 4_coloring of maximal planar graphs g9D and g12A is a path. We conjecture that there is not any other graph in all of maximal

One-way infinite hamiltonian paths in infinite maximal planar graphs

A one-way infinite Hamiltonian path is constructed in an infinite 4-connected VAP-free maximal planar graph containing one or two vertices of infinite degree. Combining this result and that of R.

A theorem on paths in planar graphs

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 [5] asserting that every

On the connectivity of maximal planar graphs

This work generalizes Etourneau's result by giving sufficient conditions in terms of the vertex degrees for G to be dp -connected.

On paths in planar graphs

This paper generalizes a theorem of Thomassen on paths in planar graphs. As a corollary, it is shown that every 4-connected planar graph has a Hamilton path between any two specified vertices x, y