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MATH 443 Problems #2 Due Tuesday October 1. These problems are for classroom presentations and some fraction of them may be final exam questions. 5. Let G be a graph with δ(G) ≥ 2, Show that G contains a cycle of length at least δ(G) + 1. 6. Let G be a graph with k components and e(G) = n(G) − k. Show that G has no cycles. 7. Let G be a 3-regular graph that has a decomposition into three perfect matchings. Prove that G has no cut edge (i.e no edge e with G\e being disconnected). 8. Let G be a semiregular bipartite graph with degrees k, l with bipartition X, Y where for each x ∈ X dG (x) = k and for each y ∈ Y dG (y) = l. Assuming k ≥ l, show that there is a matching M in G with dM (v) = 1 for all v ∈ X (said to saturate X). 9. Let G be a graph on an odd number of vertices with the property that for S ⊆ V (G), we have |S| ≥ o(G − S) − 1. Show that G has a matching that hits all but one vertex of the graph (a near perfect matching). 10. Let C1 , C2 be two cycles of G with E(C1 ) ∩ E(C2 ) 6= ∅. Show that there is a third cycle C3 in G.