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TWO CONTRIBUTIONS OF EULER
SIEMION FAJTLOWICZ. MATH 4315
Eulerian Tours. Although some mathematical problems which now can be thought of as graph-theoretical, go back to the times of Euclid, the invention of the subject is credited to Leonhard Euler (1707 - 1783), one of the most creative, versatile and prolific mathematicians of all times. Euler was more interested in explaining the ideas and patterns of his solutions than in completeness of his arguments, and by contemporary standards many of his results would not be accepted today as proofs. Nevertheless, the impact of his work was helped rather than obstructed, by this atti- tude, because it emphasized the nature of mathematical discovery. This point is very well made by Polya, and Euler is one of the most quoted mathematicians in [2]. Euler was not beyond giving attention to very simple questions and even to write papers about them. The problem which directly led to development of graph theory belongs to recreational mathematics: in Konigsburg there are seven bridges on the river Pregel, and the familiar puzzle asks whether one can take a walk around the town traversing each of the bridges exactly once. The problem was almost certainly solved before, for example by ”carefully tabulating all possible paths, therefore ascertaing, which of them, if any, meet the requirement.”, ([0], 1736.)) but Euler’s solution is much simpler and general. He stressed that his idea works for arbitrary system of rivers and islands, ([0.], 1736.) Euler represented the problem as the drawing to the right of the map below, and then noted that if the requested walk were possible, then at most two points, namely the beginning and the end of the tour, could have an odd number of arcs emanating from them.
1
A graph is a system consisting of two sets: vertices, (usually denoted by V ) and edges (denoted by E), plus an assignment of an unordered pair of vertices to each of the edges. Vertices u and v assigned to an edge e are called its endpoints, and we also say that edge e is incident with its endpoints, or that, it emanates from them. Two edges are incident if they share a common vertex. Two vertices belonging to the same edge are said to be adjacent. If the endpoints of e are equal then e is called a loop. Different edges may have the same endpoints assigned to them, and such edges are called multiple. Graphs without loops and without multiple edges are called simple. Unless it is explicitly stated all graphs considered here will have finite number of vertices and edges and often without loss of generality one can assume that they are simple. The degree of a vertex v in a loopless graph is the number edges emanating from v and (by convention) every loop contributes 2 to the degree count of its vertex. A simple graph is Eulerian if there is a sequence of vertices v 1 ..vm such that consecutive vertices are adjacent, and every edge is represented by a a unique pair of consecutive vertices. Informally, a graph is Eulerian if it can be drawn without lifting a pencil from the paper and without drawing any of the edges twice. An alternating sequence of vertices and edges of a graph is a walk if each of them (apart from the last one) is incident with the next one. Thus in the case of a simple graph to describe a walk it is enough to list its vertices. A walk without repeated vertices is called a path. If in addition to this, the last vertex is equal to the first, then the path is a cycle. An u, v-path in a graph G, is a path starting with u and ending with v. Let us define the relation | on vertices of G by putting x|y if and only if G contains a x, y-path.
E.1. | is an equivalence relation on vertices of a graph.
(E’s will denote exercises, the H’s - hints for exercises with the corresponding numbers, and the S’s - solutions.) The equivalence classes of the relation | are called components of the graph. A graph is connected if it has just one component.
S.2. The sum of di’s is twice the number of edges, because starting from sum = 0 and adding di for each vertex (of degree di), we eventually count each of the edges twice. The argument shows that the sum of degrees is twice the number of edges, and in particular, the number of edges of odd degree must be even. Indeed, the sum of odd number of odd numbers is odd.
S.3. A graph has a closed Eulerian tour if and only if it is connected and every vertex has even degree. Proceeding as in the hint, the traversed edges span a connected graph in which every vertex has even degree. Indeed, the number of traversed edges incident with the last (and thus also any other) vertex, must be even. The graph of traversed edges has an Eulerian tour starting at any given edge, because the tour can be cyclically rotated. Unless all edges were already visited, we can retrace the tour of the drawn edges starting at a vertex incident with a yet untraversed edge, and extend the tour further, [3].
The resulting algorithm is as good as they come: it is a simple search that finds a way out of blind alleys by making use of past mistakes - a trial and error method in which every error can be quickly discovered and corrected.
From Euler theorem we know that every connected graph G with at most two vertices of odd degree has an Eulerian tour T starting at one of them. Let e be an edge incident with one of the vertices of odd degree, and let G′^ be the graph obtained from G by deleting this edge. To extend e to an Eulerian tour, G′^ must be an Eulerian graph, and in particular G′^ must be connected. The last condition on G′^ is not only necessary, but it is also sufficient for completion of the Eulerian tour starting with e. This can be shown by induction on the number of edges, starting with with k2 (the complete graph with 2 vertices) as an anchor condition.
A drawback of this algorithm is that at each time we extend the partial tour by an edge must make (possibly several number) calls to a subalgorithm testing connectivity of the graph spanned by the remaining, i.e., still ”not drawn” edges of G.
S.4. Suppose that we have two vertices of odd degree. Joining them by a new edge we can reduce the problem to E.3, because the resulting graph will have a closed Eulerian tour which can be shifted to get a tour of the original graph starting at one of the vertices of odd degree. In general we can proceed by induction on ω - the the number of vertices of odd degrees (assuming that the graph is connected.) In this case, the number of times the pencil has to be lifted from the paper is ω/ 2 − 1.
S.5. The in-degree of a vertex x in a digraph D is the number of edges of the form (y, x), and the out-degree is the number of edges of the form (x, y). A necessary condition for existance of a closed Eulerian tour is that the in-degree of each vertex is equal to its out-degree. Another necessary condition is the connectivity of the graph obtained from D by ignoring the direction of edges. The algorithm described in S. can be easily modified to arbitrary digraphs.
S. 6. A graph is Hamiltonian if one can list its vertices, so that consecutive vertices are adjacent, and every vertex appears on the list exactly once. Let G be the graph whose vertices are chessboard squares, two being adjacent if and only if a knight placed on one of them can attack the other square. Then the question of existence of a knight-tour is the problem of whether the resulting graph is Hamiltonian.
A set of vertices is independent if no two of its elements are adjacent. The question what is the maximum number of mutually non-attacking queens on the chessboard is the problem of finding a largest independent set in the graph in which two squares of a chessboard are adjacent if and only if a queen placed on one of them can attack the other.
Planar Graphs. When Euler discussed the problem of touring an ”arbitrary” system of bridges he classified it as new kind of a geometry (without magnitudes) problem. Another paper of Euler which had impact on graph theory was about polyhedra, and characteristically Euler did not clearly say what he meant by this concept. A polygon is a plane area bounded by a piece-wise linear curve which does not intersect itself. A polyhedron is a solid bounded by a a finite number of polygons which are called faces. Euler stated that every polyhedron has the property that
V − E + F = 2,
where V is the number of vertices, E - edges, and F - the number of faces. A graph is planar if it can be drawn on the plane so that the arcs representing edges do not cross. A planar drawing of a graph splits the plane into disjoint regions, called also faces or countries. Two points belong to the same face if they can be connected by an arc intersecting no edges of the graph.
Theorem. The Euler formula is true for all connected planar graphs.
Proof: A connected graph without cycles is called a tree. Since the number of faces of a tree is 1, in this case, the Euler formula amounts to showing that the number of edges is one less than the number of vertices. It is easy to prove that if G is a graph in which every vertex has degree greater or equal to 2, then G must contain a cycle. Thus every tree T has a vertex of degree at most 1, and if the number of
A graph in which every vertex has degree 3 is called cubic. A fullerene is a simple planar cubic graph in which every face has 5 or 6 sides.
E.3. Prove that every fullerene has exactly 12 faces of size 5.
The girth of a graph is the size of a smallest cycle.
E.5. (optional) Show that there are cubic planar graph of girth 5 which are not fullerenes.
H. 1. If K 5 were planar then we could use Euler’s characteristic. To represent K 5 on the torus, draw this graph on a page with identified parallel sides.
H.2. Use Euler characteristic.
H.3. Express the number of edges in terms of the number of k-sided faces.
H. 4. If G contains a cycle then it contains an edge whose deletion does not disconnect the graph.
S.1. Suppose that K 5 is planar. Since every vertex has degree 4, the number of edges is 10, and thus Euler characteristic implies that the number of faces is 7. Since every face has at least 3 sides, the number of edges is at least 3F/2, which is a contradiction.
S.3. The number of edges is 1/2(5f 5 + 6f 6 ) and also 3n/2 where n is the number of vertices. Thus n = 2E/3 = 15f 5 + 18f 6. Substituting this to Euler characteristic formula and replacing the number of faces by f 5 + f 6 we get f 5 = 12.
S.4. Let G′^ be the graph obtained from G by deleting an edge contained in a cycle. Then G′^ is connected, and we can proceed by induction on the number of faces F , ( if F = 1, then G′^ is a tree, so the anchor condition holds true.) The number of faces F ′^ of G′^ is F − 1, the number of edges of G′^ is E′^ = E − 1 and the number of vertices of G′^ is V ′^ = V. Thus by induction on F
2 = V ′^ − E′^ + F ′^ = V − E + F
which proves the theorem.
E. 6 Prove or disprove that graph is planar iff it spherical.
Notation: kn,m - the complete bipartite graph denotes the graph with n + m vertices, n of which are colored red, and the remaining colored blue, so that two vertices of this graph are adjacent iff they have different colors. Complete bipartite graphs in with n = 1, are called stars.
E. Prove or disprove that k 3 , 3 is planar.
REFERENCES
- Leonhard Euler, Comm. Acad. Sci. Petrop. 8, 1736. 128 - 40. publ. 1741.
- Leonhard Euler, The Seven Bridges of Konigsburg, in ”The Word of Mathe- matics”, by James R. Newman, vol 1. p. 565 - 571.
- George Polya, Mathematical and Plausible Reasoning, Princeton University Press.
- Hugo Steinhaus, Mathematical Snapshots, Oxford University Press.
- R. Courant and H. Robins, What is Mathematics?
