By Alan Tucker

ISBN-10: 0470458380

ISBN-13: 9780470458389

This is often Alan Tuckers textbook on combinatorics and graph concept

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Extra info for Applied Combinatorics (6th Edition)

Sample text

We choose a first chord and draw it, say, outside the circle. If properly chosen, this chord will force certain other chords to be drawn inside the circle (if also placed outside the circle, they would have to cross the first chord). These inside chords will force still other chords to be drawn outside, and so on. After the first chord is drawn, the choice of placing subsequent chords inside or outside is forced. Thus, if we reach a point where a new chord will have to cross some previous chord, whether the new chord is drawn inside or outside, we can claim that the graph must be nonplanar.

There must exist even-length paths Q, Q ′ joining a with b and c, respectively (since b and c are on the left). 15b, in which Q is dashed and Q ′ is dotted. Observe that Q ′ followed by the edge (c, b) yields an odd-length path from a to b. This is impossible, since we just proved that there cannot be both an even-length path (Q) and an odd-length path (Q ′ plus (a, b)) from a to any other vertex in G. By similar reasoning, two vertices on the right cannot be adjacent. Thus, we have a bipartite arrangement of G.

17b. Our principal focus in this section is determining whether a graph is planar. We take two approaches, both based on the AC Principle. The first approach involves a systematic method for trying to draw a graph edge-by-edge with no crossing edges, in the same spirit as when we tried to determine if two graphs are isomorphic. The second approach develops some theory with a goal of finding useful properties of planar graphs. If a graph does not satisfy one or more of these properties, then we know that it cannot be planar.