## Abstract In 1976, Borodin conjectured that every planar graph has a 5‐coloring such that the union of every __k__ color classes with 1 ≤ __k__ ≤ 4 induces a (__k__—1)‐degenerate graph. We prove the existence of such a coloring using 18 colors. © 2008 Wiley Periodicals, Inc. J Graph Theory 58:139
Grünbaum's gap conjecture
✍ Scribed by Grattan Murphy; Ren Ding
- Publisher
- Elsevier Science
- Year
- 1988
- Tongue
- English
- Weight
- 550 KB
- Volume
- 69
- Category
- Article
- ISSN
- 0012-365X
No coin nor oath required. For personal study only.
✦ Synopsis
Purdy's generalization of Griinbaum's gap conjecture is roved for all arrangements w&h a sufficiently high maximum number of concurrent lines. We also improve Purdy's bounds for the general theorem and establish two lower bounds for h(A) for all arrangements.
📜 SIMILAR VOLUMES
We establish a duality principle for arrangements of pseudolines in the projective plane, and thereby prove the conjecture of Burr, Griinbaum, and Sloane that the solution T(p) of the "orchard problem" for pseudoline arrangements and the solution r(p) of the dual problem xe equa1.
## Abstract We prove that if __G__ is a triangulation of the torus and χ(__G__)≠5, then there is a 3‐coloring of the edges of __G__ so that the edges bounding every face are assigned three different colors. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 68–81, 2010
A simple proof of Grfinbaum's theorem on the 3-colourability of planar graphs having at most three 3-cycles is given, which does not employ the colouring extension. In 1958, Gr6tzsch I-5] proved that every planar graph without cycles of length three is 3-colourable. In 1963, Griinbaum [6] extended