W. T. Tutte

The Factors of Graphs

A graph G consists of a non-null set V of objects called vertices together with a set E of objects called edges, the two sets having no common element. With each edge there are associated just two vertices, called its ends. Two or more edges may have the same pair of ends.

The Dissection of Rectangles Into Squares

We consider the problem of dividing a rectangle into a finite number of non-overlapping squares, no two of which are equal. A dissection of a rectangle R into a finite number n of non-overlapping squares is called a squaring of R of order n; and the n squares are the elements of the dissection. The term “elements” is also used for the lengths of the sides of the elements. If there is more than one element and the elements are all unequal, the squaring is called perfect, and R is a perfect rectangle.

The dissection of equilateral triangles into equilateral triangles

In a previous joint paper (‘The dissection of rectangles into squares’, by R. L. Brooks, C. A. B. Smith, A. H. Stone and W. T. Tutte, Duke Math . J. 7 (1940), 312–40), hereafter referred to as (A) for brevity, it was shown that it is possible to dissect a square into smaller unequal squares in an infinite number of ways. The basis of the theory was the association with any rectangle or square dissected into squares of an electrical network obeying Kirchhoffs laws. The present paper is concerned with the similar problem of dissecting a figure into equilateral triangles. We make use of an analogue of the electrical network in which the ‘currents’ obey laws similar to but not identical with those of Kirchhoff. As a generalization of topological duality in the sphere we find that these networks occur in triplets of ‘trial networks’ N 1 , N 2 , N 3 . We find that it is impossible to dissect a triangle into unequal equilateral triangles but that a dissection is possible into triangles and rhombuses so that no two of these figures have equal sides. Most of the theorems of paper (A) are special cases of those proved below.

Toward a theory of crossing numbers

Abstract An algebraic structure, related to the crossing number, is constructed from a given graph.

On the algebraic theory of graph colorings

Abstract Some well-known coloring problems of graph theory are generalized as a single algebraic problem about chain-groups. This is transformed into a problem about the finite projective geometries over GF (2). The geometrical problem is solved up to the 5-dimensional case.

On dichromatic polynomials

Abstract A study is made of the combinatorial properties of the dichromatic polynomials of graphs, especially those properties theoretically applicable to the recursive calculation of the polynomials. The dichromatic polynomials of the complete graphs are determined and a contribution is made to the theory of chromatic roots.

On the enumeration of planar maps

A planar map is determined by a finite connected nonnull graph embedded in the 2-sphere or closed plane. It is permissible for the graph to have loops or multiple joins. It separates the remainder of the surface into a finite number of simply-connected regions called the faces of the map. We refer to the vertices and edges of the graph as the vertices and edges of the map, respectively. The valency of a vertex is the number of incident edges, loops being counted twice. A vertex-map is a planar map having exactly one vertex and no edges. Clearly a vertex-map has only one face. A map with exactly one edge is called a link-map or a loop-map according as the two ends of the edge are distinct or coincident. Thus a link-map has exactly one face and a loop-map exactly two. Two planar maps are combinatorially equivalent if there is a homeomorphism of the surface which transforms one into the other. To within a combinatorial equivalence there is only one vertex-map, one link-map and one loop-map. But the vertex-map, link-map and loop-map are combinatorially distinct from one another. Consider a planar map M which is not a vertex-map. Each face of M has an associated bounding path in the graph. We can consider this to be the path traced out by a point moving along the edges of the graph in accordance with the following rules. Normally in any small interval of time the point traces out a simple arc. On one side of this directed arc, let us say the right side, there is locally nothing but points of the face. Having started along one edge, the point continues along it, without reversing direction, until it comes to the far end. If this end is monovalent, the point then proceeds back along the same edge. This behaviour at a monovalent vertex constitutes the only exception to the rule of simple arcs in short intervals of time. The bounding path is the cyclic sequence of positions of the moving point, some of which may be repeated. We restrict it to a single cycle by the rule that it may not traverse any edge twice in the same direction. When the distinction between right and left has been made for a map, the above rules determine the bounding path of each face

Matroids and graphs

the circuits of any finite graph G define a matroid. We call this the circuitmatroid and its dual the bond-matroid of G. In the present paper we determine a necessary and sufficient condition, in terms of matroid structure, for a given matroid M to be graphic (cographic), that is the bond-matroid (circuitmatroid) of some finite graph. The condition is that M shall be regular and shall not contain, in a sense to be explained, the circuit-matroid (bondmatroid) of a Kuratowski graph, that is a graph with one of the structures shown in Figure I. Some of the intermediate results seem to be of interest in themselves. These include the theory of dual matroids in ?2, Theorem (7.3) on regular matroids, and Theorem (8.4) on the bond-matroids of graphs. Our main theorem is evidently closely related to the theorem of Kuratowski on planar graphs [1]. This states that a graph is planar (i.e., can be imbedded in the plane) if and only if it contains no graph with the point set structure of a Kuratowski graph. Indeed it is not difficult to prove Kuratowskis Theorem from ours, using the principle that a graph is planar if and only if it has a dual graph, that is if and only if its circuit-matroid is graphic. However this paper is long enough already and we refrain from adding matter not essential to the proof and understanding of the main theorem. 2. Dual matroids. We define a matroid M on a set M, its flats and their dimensions as in HI, ?1. We call the dimension of the largest flat (M) also the dimension dM of the matroid. (See HI, ?2, for the notation (S)). We write ae(S) for the number of elements of any finite set S. We proceed to give a definition of the dual of M analogous to the definition of a dual vector space in terms of orthogonality. First, in analogy with A, ?4, we define a dendroid of M as a minimal sub

On chromatic polynomials and the golden ratio

Abstract Let M be a triangulation of the 2-sphere, with k vertices. Let P(M, n) be its chromatic polynomial with respect to vertex-colorings. Then | P ( M , 1 + τ ) | ⩽ τ 5 − k where τ is the “golden ratio” ( 1 + 5 ) / 2 This result is offered as a theoretical explanation of the empirical observation that P(M, n) tends to have a zero near n=1+τ (see [1]).

On the dimension of a graph

Our purpose in this note is to present a natural geometrical definition of the dimension of a graph and to explore some of its ramifications. In