The Four Color Problem

Map Color Theorem

“…One can easily compute the genus p of the represented orientable surface. 19 13 7 2 23 9 22 10 21 3 4 5 8 15 14 11 6 20 18 16 17 13. 15 7 12 19 8 10 17 9 21 6 23 18 3 2 5 14 20 11 16 22 4 14. 1 10.…”

“…19) if n=7. Alternatively, since K7 can be embedded in the torus Sl, by adding one crosscap to the surface we obtain an embedding of K7 in N3 • Before determining the chromatic number of Klein's bottle we need to explain a graph theoretical operation.…”

Each 3-polytope with minimum degree 5 has a 7-cycle with maximum degree at most 15

“…This result was strengthened by Franklin [3] in 1922 to the existence of a vertex of degree 5 with two 6 − -neighbors. In 1940, Lebesgue [4, p. 36] gave an approximate description of the neighborhoods of vertices of degree 5 in T 5 .…”

“…The bounds w M (S 1 ) ≤ 11 [2] and w M (S 2 ) ≤ 17 [3] are tight. It was proved by Lebesgue [4, p. 36] that w M (S 3 ) ≤ 24 and w M (S 4 ) ≤ 31, which was improved much later to the tight bounds w M (S 3 ) ≤ 23 (Jendrol' and Madaras [5]) and w M (S 4 ) ≤ 30 (Borodin and Woodall [6]).…”

A new proof of 3-colorability of Eulerian triangulations

“…In 1898, Heawood [6] asserted this statement, without proof. In 1939, Franklin [3] developed relevant concepts foreshadowing the notion of nowhere-zero flows, but he did not complete a proof of Heawood's claim. Tutte [16,17] later developed the basic theory of nowherezero flows, and Steinberg [14] proved Heawood's claim in 1993 using these methods.…”