Search

In Graph theory, colouring is the best and most well-considered amplitude. The four-colour problem has contributed to great change in the world of mathematics. Graph colouring has become an essential of interest in graph theory. This often includes its disparate theoretical conclusions, its unconcluded issues and its several utilizations. Throughout this paper, the Conjecture Steinberg is studied with a counter example. Conjecture Steinberg states that "Each planar graph except cycles of length four or five is 3-colourable". Whereas an example by which there exists a planar graph with four or five cycle is 3-colourable is encountered for the Steinberg conjecture. [ABSTRACT FROM AUTHOR]

In this paper the new coloring of planar graphs, vertex-edge-face coloring, will be introduced. A vertex-edge-face coloring of a simple planar graph G is a proper coloring of all elements, including vertices, edges and faces of G. We will give two conjectures for the upper bound of vertex-edge-face and list vertex-edge-face coloring of a simple planar graph. However, we will prove these conjectures for planar graphs with a maximum degree of at least 12. [ABSTRACT FROM AUTHOR]