<P> In this map, the two regions labeled A belong to the same country, and must be the same color . This map then requires five colors, since the two A regions together are contiguous with four other regions, each of which is contiguous with all the others . A similar construction also applies if a single color is used for all bodies of water, as is usual on real maps . For maps in which more than one country may have multiple disconnected regions, six or more colors might be required . </P> <P> A simpler statement of the theorem uses graph theory . The set of regions of a map can be represented more abstractly as an undirected graph that has a vertex for each region and an edge for every pair of regions that share a boundary segment . This graph is planar (it is important to note that we are talking about the graphs that have some limitations according to the map they are transformed from only): it can be drawn in the plane without crossings by placing each vertex at an arbitrarily chosen location within the region to which it corresponds, and by drawing the edges as curves that lead without crossing within each region from the vertex location to each shared boundary point of the region . Conversely any planar graph can be formed from a map in this way . In graph - theoretic terminology, the four - color theorem states that the vertices of every planar graph can be colored with at most four colors so that no two adjacent vertices receive the same color, or for short, "every planar graph is four - colorable" (Thomas 1998, p. 849; Wilson 2014). </P> <P> As far as is known, the conjecture was first proposed on October 23, 1852 when Francis Guthrie, while trying to color the map of counties of England, noticed that only four different colors were needed . At the time, Guthrie's brother, Frederick, was a student of Augustus De Morgan (the former advisor of Francis) at University College London . Francis inquired with Frederick regarding it, who then took it to De Morgan (Francis Guthrie graduated later in 1852, and later became a professor of mathematics in South Africa). According to De Morgan: </P> <P> "A student of mine (Guthrie) asked me to day to give him a reason for a fact which I did not know was a fact--and do not yet . He says that if a figure be any how divided and the compartments differently colored so that figures with any portion of common boundary line are differently colored--four colors may be wanted but not more--the following is his case in which four colors are wanted . Query cannot a necessity for five or more be invented ..." (Wilson 2014, p. 18) </P>

Can you name the color which does not have e in it