On the Geographical Problem of the Four Colours.
By A. B. Kempe, B. A., London, England.
If we examine any ordinary map, we shall find in general a number of lines dividing it into districts, and a number of others denoting rivers, roads, etc. It frequently happens that the multiplicity of the latter lines renders it extremely difficult to distinguish the boundary lines from them. In cases where it is important that the distinction should be clearly marked, the artifice has been adopted by map-makers of painting the districts in different colours, so that the boundaries are clearly defined as the places where one colour ends and another begins; thus rendering it possible to omit the boundary lines altogether. If this clearness of definition be the sole object in view, it is obviously unnecessary that non-adjacent districts should be painted different colours; and further, none of the clearness will be lost, and the boundary lines can equally well be omitted, if districts which merely meet at one or two points be painted the same colour. (See Fig 1.)
This method of definition may of course be applied to the case of any surface which is divided into district. I shall, however, confine my investigations primarily to the case of what are known as simply or singly connected surfaces, i. e. surfaces such as a plane or sphere, which are divided into two parts by a circuit, only referring incidentally to other cases.
If, then, we take a simply connected surface divided in any manner into districts, and proceed to colour these districts so that no two adjacent districts shall be of the same colour, and if we go to work at random, first colouring as many districts as we can with one colour and then proceeding to another colour, we shall find that we require a good many different colours; but, by the use of a little care, the number may be reduced. Now, it has been stated somewhere by Professor De Morgan that it has long been known to mapmakers as a matter of experience—an experience however probably confined to comparatively simple cases—that four colours will suffice in any case. That four colours may be necessary will be at once obvious on consideration of the case of one district surrounded by three others, (see Fig. 2), but that four colours will suffice in all cases is a fact whichi is by no means obvious, and has
