By B H Korte; Jens Vygen

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Vk , ek , vk+1 be the sequence W . For i := 1 to k do: Set Wi := Euler(G, vi ). Set W := W1 , e1 , W2 , e2 , . . , Wk , ek , vk+1 . Return W . For digraphs, 2 has to be replaced by: 2 If δ + (x) = ∅ then go to 4 . Else let e ∈ δ + (x), say e = (x, y). 25. Euler’s Algorithm works correctly. Its running time is O(m + n), where n = |V (G)| and m = |E(G)|. Proof: We use induction on |E(G)|, the case E(G) = ∅ being trivial. Because of the degree conditions, vk+1 = x = v1 when 4 is executed. So at this stage W is a closed walk.

Prove that any simple undirected graph G with |E(G)| > |V (G)|−1 is con2 nected. 4. Let G be a simple undirected graph. Show that G or its complement is connected. Exercises 43 5. Prove that every simple undirected graph with more than one vertex contains two vertices that have the same degree. Prove that every tree (except a single vertex) contains at least two leaves. 6. Let G be a connected undirected graph, and let (V (G), F) be a forest in G. Prove that there is a spanning tree (V (G), T ) with F ⊆ T ⊆ E(G).

6(c)). In both cases, there are four vertices another neighbour z ∈ y, z, y , z on C, in this cyclic order, with y, y ∈ (v) and z, z ∈ (w). This implies that we have a K 3,3 minor. ✷ The proof implies quite directly that every 3-connected simple planar graph has a planar embedding where each edge is embedded by a straight line and each face, except the outer face, is convex (Exercise 27(a)). The general case of 38 2. Graphs (a) (b) (c) yi z yi yi+1 v C w v w C v w C yj z Fig. 6. 38. (Thomassen [1980]) Let G be a graph with at least five vertices which is not 3-connected and which contains neither K 5 nor K 3,3 as a minor.

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