We show the following. (1) For each integer n> 12, there exists a uniquely 3-colorable graph with n vertices and without any triangles. (2) There exist infinitely many uniquely 3-colorable regular graphs without any triangles. It follows that there exist infinitely many uniquely k-colorable regular
Uniquely colorable perfect graphs
β Scribed by Alan Tucker
- Publisher
- Elsevier Science
- Year
- 1983
- Tongue
- English
- Weight
- 883 KB
- Volume
- 44
- Category
- Article
- ISSN
- 0012-365X
No coin nor oath required. For personal study only.
π SIMILAR VOLUMES
We construct counterexamples to the conjecture of Xu (1990, J. Combin. Theory Ser. B 50, 319 320) that every uniquely r-colorable graph of order n with exactly (r&1) n&( r2 ) edges must contain a K r . ## 2001 Academic Press Harary et al. [4] constructed uniquely r-colorable graphs containing no
## Abstract The generalized Petersen graph __P__(6__k__ + 3, 2) has exactly 3 Hamiltonian cycles for __k__ β₯ 0, but for __k__ β₯ 2 is not uniquely edge colorable. This disproves a conjecture of Greenwell and Kronk [1].
We consider the question of the computational complexity of coloring perfect graphs with some precolored vertices. It is well known that a perfect graph can be colored optimally in polynomial time. Our results give a sharp border between the polynomial and NP-complete instances, when precolored vert