Notes on grading

The well-known theorem of Erdos and Posa says that a graph G has either k vertex-disjoint cycles or a vertex set X of order at most f(k) such that G@?X is a forest. Starting with this result, there are many results concerning packing and covering cycles ...

We prove that for any given @e>0 and d@?[0,1], every sufficiently large (@e,d)-dense graph G contains for each odd integer r at least (d^r-@e)|V(G)|^r cycles of length r. Here, being (@e,d)-dense means that every set X containing at least @e|V(G)| ...

The class of doubly chordal graphs, which is a subclass of chordal graphs and a superclass of strongly chordal graphs, arises in many application areas. Many NP-complete optimization problems on chordal graphs like domination and Steiner tree can be ...