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“think” — 2017/9/4 — 17:46 — page 255 — #265
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11.4. EULER’S BRIDGES
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Figure 11.3: The graph representing the seven bridges of K¨onigsberg. The figure shows the land mass and the two islands. The seven bridges are also clearly depicted by heavy segments. It had been a problem of some standing to determine whether it was possible to walk a path that crossed each bridge exactly once. The only rule here is that, once a bridge is entered, one must walk all the way across it. And one cannot walk back. You can only cross each bridge once. Euler’s profound insight was that one should construct from this problem a graph. Here we are not talking about the graph of a function but rather about a combinatorial graph. One accomplishes this task by placing a vertex or node in each land mass and then connecting two vertices with an edge if the two corresponding land masses are connected by a bridge. See Figure 11.3. Now the problem becomes Can one traverse the graph shown in Figure 11.3 along a path in such a fashion that each edge is traveled once and only once? What Euler noticed was that, except for the node where the path begins and the node where the path ends, each of the nodes is entered and exited the same number of times. One does not stop at a node. One passes through it. Therefore, of the four nodes, three of them must be the terminus of an even number of edges. But such is not the case. In fact all four of the nodes has odd valence—meaning that an odd number of edges terminates there. Thus the desired path is impossible. The problem changes if we demand that the path begin and end at the same place. For then that beginning/ending node will also have even valence. In fact all the nodes will have to have even valence. But such is not the case.
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