(a,b)-decomposability

In graph theory, the (a, b)-decomposability of an undirected graph is the existence of a partition of its edges into a + 1 sets, each one of them inducing a forest, except one who induces a graph with maximum degree b. If this graph is also a forest, then we call this a F(a, b)-decomposition.

A graph with arboricity a is (a, 0)-decomposable. Every (a, 0)-decomposition or (a, 1)-decomposition is a F(a, 0)-decomposition or a F(a, 1)-decomposition respectively.

Graph Classes

Notes

  1. ↑ Gonçalves (2009), conjectured by Balogh et al. (2005). Improving results by Nash-Williams (1964) then Balogh et al. (2005).
  2. ↑ 2.0 2.1 Implied by Nash-Williams (1964).
  3. ↑ He et al. (2002)
  4. ↑ Implied by Montassier et al. (2012), improving results by He et al. (2002), then Kleitman (2008).
  5. ↑ Independently proved by Yang & Zhang (2011) and implied by Montassier et al. (2012), improving results by He et al. (2002) for girth 11, then Bassa et al. (2010) for girth 10 and Borodin et al. (2008a) for girth 9.
  6. ↑ Borodin et al. (2009b), even if not explicitly stated.
  7. ↑ Borodin et al. (2009a), improving results by He et al. (2002), then Borodin et al. (2008b).
  8. ↑ Proved without explicit reference by Guan & Zhu (1999).

References (chronological order)