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With the help of Neutrosophy and Quad-stage Method, the proof for negation of “the four color theo-rem” is given. In which the… Expand In this paper, we will define the necessary concepts to formulate map coloring problems. Then, we will prove Euler’s formula and… Expand A formal proof has not been found for the four color theorem since 1852 when Francis Guthrie first conjectured the four color… Expand We prove τodd(G) ≤ 2Vodd(G) for each planar graph G where vodd(G) is the maximum number of edge-disjoint odd cycles and τodd(G… Expand Based on a representation of primitive proof objects as λ-terms, which has been built into the theorem prover Isabelle recently… Expand In this paper, we introduce graph theory, and discuss the Four Color Theorem. Then we prove several theorems, including Euler’s… Expand Abstract In this note, we show that the edges and faces of any plane graph with maximum degree three can be simultaneously… Expand Let f be a face of a plane graph G. The Three and Five Color Theorem proved here states that the vertices of G can be colored… Expand