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Average degree conditions forcing a minor
Mader first proved that high average degree forces a given graph as a minor.
Often motivated by Hadwiger's Conjecture, much research has focused on the
average degree required to force a complete graph as a minor. Subsequently,
various authors have consider the average degree required to force an arbitrary
graph as a minor. Here, we strengthen (under certain conditions) a recent
result by Reed and Wood, giving better bounds on the average degree required to
force an -minor when is a sparse graph with many high degree vertices.
This solves an open problem of Reed and Wood, and also generalises (to within a
constant factor) known results when is an unbalanced complete bipartite
graph
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