2. Region(face) colourings Definitions 46: A edge of the graph is called a bridg if the edge is not in any circuit. a connected planar graph is called a map, If the graph has not any bridge Definition 47: A proper region coloring of a map g is an assignment of colors to the region of G, one color to each region, such that adiacent regions receive different colors. An proper region coloring in which k colors are used is a k-region coloring. A map g is k-region colorable if there exists an s coloring of g for some s sk. The minimum integer k for which G is k- region colorable is called the region chromatic number. We denoted by x (G). If x (G) k, then g is k-region chromatic
▪ 2. Region(face) colourings ▪ Definitions 46: A edge of the graph is called a bridge, if the edge is not in any circuit. A connected planar graph is called a map, If the graph has not any bridge. ▪ Definition 47: A proper region coloring of a map G is an assignment of colors to the region of G, one color to each region, such that adjacent regions receive different colors. An proper region coloring in which k colors are used is a k-region coloring. A map G is k-region colorable if there exists an scoloring of G for some s k. The minimum integer k for which G is k- region colorable is called the region chromatic number. We denoted by *(G). If *(G) = k, then G is k-region chromatic
Four Colour Conjecture Every map(plane graph) is 4-region colourable Definition 48: Let g be a connected plane graph. Construct a dual gd as follows: PLace a vertex in each region of G; this forms the vertex set of gd. 2)Join two vertices of Gd by an edge for each edge common to the boundaries of the two corresponding regions of G 3)Add a loop at a vertex v of Gd for each bridge that belongs to the corresponding region of G. moreover, each edge of ga is drawn to cross the associated edge of G, but no other edge of g or gd
▪ Four Colour Conjecture Every map (plane graph) is 4-region colourable. ▪ Definition 48：Let G be a connected plane graph. Construct a dual Gd as follows: ▪ 1)Place a vertex in each region of G; this forms the vertex set of Gd . ▪ 2)Join two vertices of Gd by an edge for each edge common to the boundaries of the two corresponding regions of G. ▪ 3)Add a loop at a vertex v of Gd for each bridge that belongs to the corresponding region of G. Moreover, each edge of Gd is drawn to cross the associated edge of G, but no other edge of G or Gd
