the contest problem book iv annual high school examinations, 1973-1982

[...]... GRAPHS AND THEIR USES The Problem of the Bridges of Konigsberg The theory of graphs is one of the few fields of mathematics with a definite birth date The first paper relating to graphs was written by the Swiss mathematician Leonhard Euler (1707-1783), and it appeared in the 1736 volume of the publications of the Academy of Sciences in St Petersburg (Leningrad) Euler (pronounced 'oyler') is one of the most... nature of the land and the climate are such that one or another of the wells frequently runs dry; it is therefore important that the people of each house have access to each of the three wells After a while, the residents a, b and c develop rather strong dislikes for one another and decide to construct paths to the three wells x, y, z in such a manner that they avoid meeting each other on their way... geometric view, it is The difficulty lies in the precise definition of "curve", which we omit here, together with the proof of the Jordan curve theorem You may take the theorem as an evident fact This theorem implies the intuitively obvious result that if any two points on the closed curve C, say a and y, are connected by a curve ay which has no other points in common with C, then, except for its end... occurred together in the same tomb then their time periods must have overlapped, they constructed a graph in which the vertices correspond to the artifacts and the edges correspond to pairs of artifacts which appeared together in some tomb By representing this graph as an interval graph and interpreting the intervals as time periods during which the artifact was in use, they were then able to arrange the tombs... the complement of the graph G in Figure 1.1, and it is customary to denote it by G If we take the complement of G, we get back to G; together the edges in the two graphs G and G make up the complete graph connecting their vertices Problem Set 1.2 1 Draw the complement of the graph in Figure 1.2 2 How many edges have the complete graphs Ks, K6 and K/! 3 Express in terms of n the number of edges in the. .. 1.8 are isomorphic in spite of the fact that they have been drawn in different manners (The term "isomorphic" is a much used one in mathematics; it is derived from the Greek words iso- the same, and morphe-form.) Often one is faced with the problem of deciding whether two graphs are isomorphic At times there are obvious reasons why this cannot be the case For example, the graphs in Figure 1.9 cannot... theory As the number of vertices increases, the number of ways of naming them grows very fast, and isomorphisms between the graphs become very hard to find, even with a computer Problem Set 1.3 1 Show that the graphs in Figure 1.1, Figure 1.2 and Figure 1.6 are not isomorphic to each other 2 Give another reason why the two graphs in Figure 1.11 cannot be isomorphic 3 Name the vertices in the two graphs... formulas is not always the best criterion for judging the depth of a mathematical theory We can also indicate an application of planar graphs to an eminently practical problem In addition to the previous interpretations, a graph can be thought of as the diagram for an electrical network, with the edges representing the conducting wires connecting the various junctions One of the most effective ways of mass... whether a sum of integers is odd or even, we can disregard the even terms; the sum is even or odd depending upon GRAPHS AND THEIR USES 20 whether it contains an even or an odd number of odd summands When we apply this observation to the fact that the sum of the degrees is even, we arrive at the following result: THEOREM 1.1 A graph has an even number of odd vertices (We include in this statement the. .. where there are no odd vertices, since 0 is an even number.) There are special graphs in which all degrees are the same: The graph is then called regular of degree r and, according to the handshaking lemma, the number of its edges is m = tnr, where n is the number of vertices The graphs in Figure 1.22 are regular of degree 3 and 4, respectively Figure 1 22 In the complete graph K n with n vertices there . audience of high school students and laymen. Most of the volumes in the New Mathematical Library cover topics not usually included in the high school curricu- lum; they vary. 29 The Contest Problem Book IV Annual High Scbool Matbemaucs Exanunations 1973-1982 Compiled and witb solutions by R. A. Artino, A. M Gaglione and N. Shell 30 The Role. Mathematics by Ross Honsberger 24 Geometric Transformations III by I. M. Yaglom, translated by A. Shenitzer 25 The Contest Problem Book III Annual High Scbool Mathematics