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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 topological spaces, which are branches of pure mathematics, are difficult to find, the significance of this paper is to find some in graph theory via some topological space concepts, we introduced some well-known concepts in topology and related them to a graph such as (graph interior, graph closure, graph exterior, graph boundary and graph limit point). We study a characterization and relationships between them and present some results and theorems about this notions. In this work includes all Species graphs. [ABSTRACT FROM AUTHOR]

Graph theory is a branch of mathematics which is quit rich and interesting. The study of domination is one of the fastest growing area in graph. There are different types of domination such as connected domination, independent domination, total domination, paired domination, secure domination , etc. "The secure dominating set S of a graph G is a dominating set with the property that each vertex u∈V\ S is adjacent to a vertex v∈S such that (S\ v) Uu is a dominating set. The minimum cardinality of such a set is called the secure domination number and denoted by γs(G)". In this paper we discuss about the secure domination for some special graphs. [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]