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By Herbert S. Wilf

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2. Let G be a path of n vertices. What is the size of the largest independent set of vertices in V (G)? 3. Let G be a connected graph in which every vertex has degree 2. What must such a graph consist of? Prove. 4. Let G be a connected graph in which every vertex has degree ≤ 2. What must such a graph look like? 5. Let G be a not-necessarily-connected graph in which every vertex has degree ≤ 2. What must such a graph look like? What is the size of the largest independent set of vertices in such a graph?

Indeed, if not, then we arrived at v one more time than we departed from it, each time using a new edge, and finding no edges remaining at the end. Thus there were an odd number of edges of G incident with v , a contradiction. Hence we are indeed back at our starting point when the walk terminates. Let W denote the sequence of edges along which we have so far walked. If W includes all edges of G then we have found an Euler tour and we are finished. Else there are edges of G that are not in W . Erase all edges of W from G, thereby obtaining a (possibly disconnected multi-) graph G .

2(a) has one and the graph in Fig. 2(b) doesn’t. Fig. 2(a) Fig. 2(b) * Did you realize that the number of people who shook hands an odd number of times yesterday is an even number of people? 25 Chapter 1: Mathematical Preliminaries Fig. 3(a) Fig. 3(b) Likewise, not every graph has an Eulerian path. The graph in Fig. 3(a) has one and the graph in Fig. 3(b) doesn’t. There is a world of difference between Eulerian and Hamiltonian paths, however. If a graph G is given, then thanks to the following elegant theorem of Euler, it is quite easy to decide whether or not G has an Eulerian path.

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