We prove that for every k there exists d = d(k) such that every graph of average degree at least d contains a subgraph of average degree at least k and girth at least six. This settles a special case of a conjecture of Thomassen.

An (I,Fd)-partition of a graph is a partition of the vertices of the graph into two sets I and F , such that I is an independent set and F induces a forest of maximum degree at most d. We show that for all M < 3 and d ≥ 2 3−M − 2, if a graph has maximum average degree less than M , then it has an (I,Fd)-partition. Additionally, we prove that for all 8 3 ≤ M < 3 and d ≥ 1 3−M , if a graph has ma...

In unweighted case, approximation ratio for the independent set problem has been analyzed in terms of the graph parameters, such as the number of vertices, maximum degree, and average degree. In weighted case, no corresponding results are given for average degree. It is not appropriate that we analyze weighted independent set algorithms in terms of average degree, since inserting the vertices w...