Some Historical and Intuitive Aspects of Graph Theory

IN THIS INTRODUCTORY SURVEY of a small portion of graph theory, the deep and often subtle combinatorial reasoning is deliberately omitted. Here are the names of seven mathematicians together with the graphical topics most closely associated with each. Names of some of the other mathematicians who have concentrated on these topics follow parenthetically. The order is approximately chronological but will not be followed exactly in the discussion. 1. Euler K6nigsberg bridge problem. 2. Cayley Trees (Hamilton, Jordan, Kirchhoff, Sylvester). 3. Heawood Four color conjecture (Kempe, Coxeter). 4. Kuratowski Planar graphs (Fary, Whitney, MacLane). 5. Menger Separation theorem. 6. Frucht The group of a graph. 7. Whitney Line-graphs (Krausz). We must start with Euler, 1707-83. Euler can be truly regarded as the originator not only of graph theory, but simultaneously of the field of mathematics known as topology. See Tucker and Bailey [6] for an interesting exposition of topology. He started with a famous unsolved problem of his day called the Konigsberg bridge problem. The same city, now called Kaliningrad, is located on the Pregel River. The area in question contains two islands linked to each other and to the banks of the river by seven bridges as in Figure 1. The problem was to start at any one of the four land areas and without swimming or flying or traveling around the world to traverse each bridge exactly once and return to the starting point. One can see immediately that there are many ways of trying this problem without solving it. The tremendous contribution of Euler in this case was negative, for he proved that the problem is unsolvable. There is a beautiful article giving Euler's reasoning on this problem; Euler [3]. We mention here only the formulation, rather than the details. Replace each land area by a point or vertex and replace each bridge by a line or edge joining the points corresponding to these land areas. The result is a collection of points together with lines joining them: Figure 2. Such a configuration is now called a graph. We have labeled the points a, b, c, and d corresponding to the land areas of Figure 1. At each point of Figure 2, we have written in parentheses the number of lines incident to this point; this number is the degree of the point. A point of a graph is even (odd) if its degree is even (odd). EULER'S THEOREM. A connected graph can be traversed by a complete closed line sequence if and only if every point is even.