MATH 5707 Midterm 2
نویسنده
چکیده
Proof. Let n = |V |, and assume without loss of generality that |E| = n (else we could remove edges from G until |E| = n and apply the argument). Also assume WLOG that V ∩ E = ∅ (else, we can rename the vertices). Note that we must have n ≥ 3, since a simple graph with 2 or fewer vertices can have at most 1 edge. Now we seek a bijection f : V → E with each v ∈ V satisfying v / ∈ f(v). Construct a bipartite graph (G̃;V,E), where v ∈ V is adjacent to e ∈ E if and only if v / ∈ e. We seek a perfect matching, or simply an E-complete matching of G̃. First we note that since each e ∈ E has exactly 2 endpoints in G, it has exactly two elements of V which are not neighbors of e in G̃. That is, for each e ∈ E, we have degG̃(e) = n−2. In other words, |N({e})| = n−2 ≥ 1, since n ≥ 3. Thus, we have the Hall condition satisfied for subsets of E of size 1. Furthermore, we have that any nonempty subset P of E satisfies |N(P )| ≥ n− 2 and thus we have verified the Hall condition for all subsets of E with size ≤ n− 2. It remains to verify the Hall condition for subsets P of E with |P | = n− 1 and |P | = n. If |P | = n, then P = E, and we claim that N(P ) = V . Indeed, assume the contrary. Then there exists v ∈ V such that v is isolated
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