Domination number of cubic graphs with large girth
✍ Scribed by Daniel Král'; Petr Škoda; Jan Volec
- Publisher
- John Wiley and Sons
- Year
- 2011
- Tongue
- English
- Weight
- 219 KB
- Volume
- 69
- Category
- Article
- ISSN
- 0364-9024
No coin nor oath required. For personal study only.
✦ Synopsis
We show that every n-vertex cubic graph with girth at least g have domination number at most 0.299871n+O(n / g) < 3n / 10+O(n / g) This research was done when the Petr Škoda was a student of
📜 SIMILAR VOLUMES
It was proved by Hell and Zhu that, if G is a series-parallel graph of girth at least 2 (3k -1)/2 , then χ c (G) ≤ 4k/(2k -1). In this article, we prove that the girth requirement is sharp, i.e., for any k ≥ 2, there is a series-parallel graph G of girth 2 (3k -1)/2 -1 such that χ c (G) > 4k/(2k -1)
Suppose G and H are graphs. We say G is H-colorable if there is a homomorphism (edge-preserving vertex mapping) from G to H. We say a graph G is uniquely H-colorable if there is an onto homomorphism c from G to H, and any other homomorphism from G to H is the composition o o c of c with an automorph
## Abstract The numbers of unlabeled cubic graphs on __p = 2n__ points have been found by two different counting methods, the best of which has given values for __p ≦__ 40.
It is proved that if G is a planar graph with total (vertex-edge) chromatic number χ , maximum degree and girth g, then χ = + 1 if ≥ 5 and g ≥ 5, or ≥ 4 and g ≥ 6, or ≥ 3 and g ≥ 10. These results hold also for graphs in the projective plane, torus and Klein bottle.