1) Show that if G has two edge-disjoint spanning trees, then it has a spanning E

Question

1) Show that if G has two edge-disjoint spanning trees, then it has a spanning Eulerian subgraph. 2) Show that if G is 2k-regular for some k  1, then it has a spanning 2-regular subgraph. (You may assume without loss of generality that G is connected.) (Hint. Reduce this to a matching problem in a bipartite graph H = (X U Y;E') as follows. For each vertex v, create two copies v+ in X and v- in Y . Now use an Euler tour of G to define E' so that (i) H is k-regular and (ii) a perfect matching in H yields a 2-regular subgraph in G.)

Explanation / Answer

www.math.ku.edu/../HWanswers725-8.pdf

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