Proof of Four Color Map Theorem by Using PRN of Graph

This paper intends to study the relation between PRN and chromatic number of planar graphs. In this regard we investigate that isomorphic or 1 isomorphic graph may or may not have equal PRN and few other related results. Precisely, we give simple proof of Four Color Map Theorem.


INTRODUCTION
In this section, we present a brief survey of those results of graph theory, which we shall need later. The reader is referred to [6,7,8,10] for a fuller treatment of the subject.

Graphs:
A graph G is an ordered pair (V (G), E (G) ) where i) V(G) is a non empty finite set of elements , known as vertices. V (G) is known as vertex set. ii) E(G) is a family of unordered pairs ( not necessarily distinct ) of elements of V, known as edges of G. E(G) is known as Edge set.
Each vertex of graph G is represented by a point or small circle in the plane. Every edge is represented by a continuous arc or straight line segment. A certain pairs of vertices of graph are joined by two or more edges, such edges are known as multiple or parallel edges. An edge joining a vertex to itself, is called a loop. A graph without loops or multiple edges is called a simple graph. Non-simple graphs are known as multiple graphs. The degree or valency of a vertex v of graph G is the number of edges incident at that v. It is denoted by d (v). A vertex of degree 1 is called a pendent vertex. A vertex of zero degree is said to be isolated vertex. An edge, whose one end vertex is a pendant vertex, is known as pendant edge.
A walk of a graph is defined as a finite alternating sequence of vertices and edges, beginning and ending with vertices, such that each edge is incident with the vertices preceding and following it. Vertices of graph with which a walk begins and ends, are called its terminal vertices. A walk, in which terminal vertices are same, is called as the closed walk .Otherwise open walk. A graph G is said to be the connected graph if there exists at least one walk between every pair of vertices in G .Otherwise graph G is disconnected. The vertex connectivity of a connected graph G is defined as the minimum number of vertices whose removal from G leaves the remaining graph disconnected. A connected graph is said to be 2-connected if its vertex connectivity is two. [7] 1.2 Fat K n : A complete graph Kn is said to be fat complete graph on n vertices (Fat K n ) if there exists at least one parallel edge between vertices of Kn.

Isomorphism:
A graph G 1 (V 1 , E 1 ) is said to be isomorphic to the graph G2(V 2 , E 2 ) if i) There is a one to one correspondence between the vertex sets V 1 and V 2 . ii) There is a one to one correspondence between the edge sets E1and E 2 in such a way that if e1 is an edge with end vertices u 1 and v 1 in G 1 then the corresponding edge e 2 in G 2 has its end points the vertices u 2 and v 2 in G 2 which correspond to u 1 & v 1 respectively. Such a pair of correspondence is called a graph isomorphism.

Separable graph:
A connected graph G is said to be separable if its vertex connectivity is one. All other connected graphs are called nonseparable graphs. A connected graph G is said to be separable if there exists a subgraph G 1 in G such that G(the complement of G 1 in G) and G1 have only one vertex in common. 1.5 1-isomorphism: Two graphs G 1 and G 2 are said to be 1-isomorphic graphs if they become isomorphic to each other under repeated application of the following operation 1. Operation1: Split a cut vertex of graph into two vertices to produce two disjoint subgraphs. Therefore, two graphs G 1 and G 2 are said to be 1-isomorphic graphs if and only if the blocks or components of the graph G 1 are isomorphic to the blocks or components of the graph G 2 . From this definition, it is clear that two non-separable graphs are 1 isomorphic iff they are isomorphic. [7] 2. PLANAR GRAPHS 2.1 Planar Graph: A graph G is a planar graph if it is possible to represent it in the plane such that no two edges of the graph intersect except possibly at a vertex to which they are both incident. Any such drawing of planar graph G in a plane is a planar embedding of G.
If x any point in the plane of a planar graph that is neither a vertex nor a point on an edge, the set of all points in the plane that can be reached from x by traversing along a curve that does not have a vertex of the graph or a point of an edge as an intermediate point, is the region of the graph that contains x. Thus the plane graph G partitions the plane into the different regions of G. Among these regions there is exactly one region whose area is not finite, is called exterior or infinite region. Every other region is an interior region. The boundary of a region is a sub-graph formed by the vertices and edges encompassing that region. If the boundary of the exterior region of a planar graph is a cycle, that cycle is known as the maximal cycle of that graph. The degree of the region is the number of edges in a closed walk that encloses it. The region formed by three edges is known as triangular region. The region formed by four edges is known as rectangular region. [6,7] Theorem 2.1: If a connected planar graph of n vertices, m edges has f regions or faces, then n -m + f = 2 2.2 Geometric Dual: Let G be a plane graph with n Regions or faces say R 1 , R 2 , R 3 , . . .R n .Let us place points ( say vertices ) V 1 , V 2 , V 3 , . . . V n , one in each of the regions. Let us join these vertices V i according to the following procedure. i) If two regions Ri and Rj are adjacent then draw a line joining vertices Vi and Vj that intersect the common edge between Ri and Rj exactly once. ii) If there are two or more edges common between Ri and Rj , then draw one line between vertices Vi and Vj for each of the common edges. iii) For an edge 'e' lying entirely in one region say Ri, draw a self loop at pendant vertex Vi intersecting e exactly once. By this procedure, we obtain a new graph G* consisting of V 1 , V 2 , V 3 , . . . V n vertices and edges joining these vertices. Such a graph G* is called a geometric dual of G (a dual of G). [7] Theorem 2.2: The geometric dual of a planar graph is planar.

* isomorphism:
Two graphs are said to be *isomorphic if their geometric duals are isomorphic.
Every graph is * isomorphic to itself. The definition is symmetric, and if G 1 and G 2 are *isomorphic to G 2 and G 3 , respectively, then G 1 and G 3 are * isomorphic. Thus *isomorphism is an equivalence relation. Suppose two graphs are * isomorphic, then it is clear that the two graphs must have same

The Bulletin of Society for Mathematical Services and Standards Vol. 11
number of edges and same number of regions formed by equal number of edges. There is no any condition on number of vertices of the two graphs. Thus these conditions are necessary for two graphs to be *isomorphic. However, these conditions are not sufficient. [4]

Four Color Map Problem:
Planar map is a set of pairwise disjoint subsets of the plane, known as regions of the map. Two regions a map are adjacent if they have a common boundary that is not a corner. A vertex or point of a map is said to be corner if it is a common point of three or more regions. A coloring of a graph is an assignment of colors to its vertices (or regions) so that no two adjacent vertices (or regions) have the same color. The set of all vertices (or regions) with same color in graph, is called a color class. 2.6 HB Graph: A planar graph is said to be HB graph if it has a pivot region. Actually such graphs should name as Neighborhood graphs, but for convenience, we select two continue letters from that name. A planar graph with only one pivot region is known as Uni-HB graph. A graph having two or more pivot regions is called Multi-HB graph. A HB graph, in which every region is a pivot region of that graph, is called Complete HB graph. The Wheel graph on 5 vertices is an Uni-HB graph. The complete graphs on vertices 1,2,3,4 are complete HB graphs. [5] 3. THE MAIN RESULTS Theorem 1: If G is any planar graph then PRN (G) is at most 4. Proof: The graphs K 1 , K 3 , K 4 -{e}, and K 4 have respectively PRN 1, 2, 3 and 4. Suppose graph G has PRN 5. So there exist at least five regions which are adjacent to each other. Let us place one vertex in each region. In the geometric dual of G, at least five vertices are adjacent to each other, which form a complete graph on 5 vertices, K 5 . So K 5 is a subgraph of G*. But graph K 5 is not planar graph. Therefore graph G* is not planar graph, which is contradiction. Hence the proof.