Chandra Chekuri Scribe : Quan Geng 1
نویسنده
چکیده
Matroid union and matroid intersection are closely related in the sense that one can be derived from the other. However they are from different perspectives and have different applications. To motivate matroid union theorem we state a well known theorem of Tutte and Nash-Williams on packing disjoint spanning trees in graphs. Theorem 1 (Nash-Williams and Tutte) An undirected multi-graph G = (V, E) contains k edge-disjoint spanning trees iff for every partition P of V into sets, V 1 , V 2 ,. .. , V , the number of edges crossing the partition P is at least k(− 1). It is easy to see that the condition is necessary; if T 1 ,. .. , T k are the edge-disjoint spanning trees then each T i has to contain at least − 1 edges across the partition P to connect them. A useful corollary of the above was observed by Gusfield. It is an easy exercise to derive this from the above theorem.
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