Undirected graphs and networks

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166 General Mathematics 2 The simplified map of Africa below shows borders between countries. Print it from your Maths Quest CD and colour it using the minimum number of colours. Africa – Political Azimuthal Equal Area Projection 3 Suggest the minimum number of colours needed to colour any map. N 0 500 1000 km
Eulerian paths and circuits Chapter 21 Undirected graphs and networks 167 A path is a series of vertices connected by edges: it displays how B to travel from one vertex to another via an edge. Just as in everyday A Fig 61 life a path is something we can walk along, we may consider a E path in a graph as something that can be traversed. In a path the C D vertices are used only once and the edges are traversed (passed over) only once; that is, edges and vertices listed in a path may not be repeated. Some examples of paths in the graph above are: ABD, ABDE and ACDB. The sequence ABCA is not a path since there is no edge which connects vertex B to vertex C. A circuit (or cycle) is a path which starts and finishes at the same vertex and no edge is traversed (passed over) more than once. An example of a circuit from the above graph is ABDCA. 1. A path is a series of vertices connected by edges. 2. A circuit (or cycle) is a path which starts and finishes at the same vertex and no edge is traversed more than once. An Eulerian path is a path which uses each edge in a graph only once, however a vertex may be repeated. For the graph at right, an Eulerian path could start at A, travel to C–D–A–B–D then finish at E. Another Eulerian path is E–D–C–A–B–D–A. Do you remember a children’s game in which you must draw a picture of an envelope (or a house) without lifting your pencil off the paper and without going over any line twice? This is an example of an Eulerian path. A graph may have more than one Eulerian path or it may not be possible to draw any Eulerian paths. An example of a graph for which an Eulerian path is not possible is shown at right. A B Fig 62 C D If an Eulerian path starts and finishes at the same vertex, it is 2 called an Eulerian circuit. Therefore, an Eulerian circuit is a 1 3 path which starts and finishes at the same vertex, uses all the edges and does not go over any edge twice. An Eulerian circuit 4 5 may be drawn for the graph at right. The Eulerian circuit is 1–2–3–5–4–3–1. Eulerian paths and circuits have many real-life applications in areas such as planning delivery routes and communication networks. Consider the following example of a practical application of an Eulerian circuit. Postal workers collect their mail from the distribution centre, deliver mail along their particular route where each street (edge) is crossed once. At the end of their delivery route, the postal workers return to the distribution centre. Other examples include recycling and garbage collection, newspaper and junk mail deliveries and so on. E
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Undirected graphs and networks

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