3-Coloring Of Graphs Without Assigned Faces

In this article, we give a new form of Steinberg's conjecture by showing that every plane graph without i(4≤i≤6) faces in which any two triangles have distance at least 3 is 3-colorable. We also get another expression of Steinberg's conjecture about plane graphs, that is, every plane graph without i(4≤i≤5) faces in which any two triangles have distance at least 3 and no 3-face is adjacent to any 6-faces is 3-colorable. We also get the following result, all plane graphs without i(4≤i≤7) faces in which any two triangles have distance at least 2 are 3-colorable. After that we extend Steinberg's conjecture by showing that all simple graphs embedded in the surface of nonnegative characteristic number without i(4≤i≤6) faces in which any two triangles have distance at least 3 are 3-colorable. Finally, we get the other two claims for simple graphs embedded in the surface of nonnegative characteristic number.