On covering an independent set in a grid with a second independent set
โ Scribed by Rue, Rachel
- Publisher
- John Wiley and Sons
- Year
- 1996
- Tongue
- English
- Weight
- 1011 KB
- Volume
- 23
- Category
- Article
- ISSN
- 0364-9024
No coin nor oath required. For personal study only.
โฆ Synopsis
We show that for every independent set 0 in an n x m grid, n, m > 1, there is a second independent set X with the property that every member of 0 is adjacent to a t least one member of X. The proof gives a construction for X. Equivalently, we show that for every maximal independent set in a grid, there is a second, disjoint, maximal independent set.
๐ SIMILAR VOLUMES
For integers m, n โฅ 2, let f (m, n) be the minimum order of a graph where every vertex belongs to both a clique of cardinality m and an independent set of cardinality n. We show that f (m, n) = ( โ m -1 + โ n -1) 2 .
In 1975, John Sheehan conjectured that every Hamiltonian 4-regular graph has a second Hamiltonian cycle. Combined with earlier results this would imply that every Hamiltonian r-regular graph (r 3) has a second Hamiltonian cycle. We shall verify this for r 300.
We prove the conjecture made by O. V. Borodin in 1976 that the vertex set of every planar graph can be decomposed into an independent set and a set inducing a 3-degenerate graph.