5 Color Theorem in Graph Theory

The graph is a data structure that is used extensively in real-life.

Planar Graph: If a graph can be drawn on the plane without crossing, it is said to be planar. Coloring of a simple graph is the assignment of color to each vertex of the graph so that no two adjacent vertices are assigned the same color.

Bi-Partite Graphs: A bipartite graph, also known as a bi-graph, is a set of graph vertices that has been dissected into two distinct sets, with no graph vertices in the same set being adjacent. A bipartite graph is a k-partite graph with k=2 as a particular instance.

Chromatic Number: The minimum number of colors needed to paint a graph G is called the chromatic number of G & is denoted by – μ (G).

Adjacent Regions: An assignment of colors to the regions of a map such that adjacent regions have different colors. A map ‘M’ is n – colorable if there exists a coloring of M which uses ‘n’ colors.

Five Color Theorem 

Theorem 1: Any finite planar graph G can be colored using five colors (or fewer).

Proof:

To prove the theorem, we use induction on the number of vertices of the graph. Let G be a planar graph with m vertices. Since G is a planar graph and connected, hence there exists at least one vertex of degree less than or equal to five.

Let v be the vertex of degree≤5, and consider the induced subgraph G’ of G on G-v. According to the induction hypothesis, induced subgraph G’ requires no more than five colors. If the degree of vertex v is less than 4, then there is no difficulty in proper coloring. But the vertex v has exactly five neighbors and they are all differently colored as shown in the figure.

Suppose there is a path between vertices a and c, say a->v1->v2->…->vn->c. Color alternatively a by p1, v1 by p2, v2 by p1, and so on. Also, according to the property of the planar graph, there is no similar path between b and d (because it will intersect the path a and c). Hence there is no path of alternating colors p2 and p4 through vertices b and d. So, vertex d can be painted with the color p2 and vertex b is still with the color p2. Now we have color p4 left over with which we can paint vertex v.

Similarly, we can assume that there is no path between a and c and a path between b and d, then we can paint v with the color p1 or p3, instead of p4. Hence the theorem.

Theorem 2: If G is a simple graph with the largest vertex degree Δ, then G is (Δ+1) colorable.

Proof:

We prove the theorem by induction on the number of vertices. Suppose G is a simple graph with m vertices. If we delete the vertex v and its incident edges, then the graph will remain a simple graph with m-1 vertices and the largest degree at most Δ. By our induction hypothesis, this graph is (Δ+1) colorable. Now, color the vertex v with different colors from the vertices adjacent to v for (Δ+1) coloring of graphs G.

Previous Year GATE Question

 Let G be a simple, finite, undirected graph with vertex set {v1,v2,v3,…,vn}. Let Δ(G) denote the maximum degree of G and let N={1,2,…} denote the set of all possible colors. Color the vertices using the following greedy strategy: 

for i=1, . . . , n

color(Vi)<-min{j ∈ N : no neighbor of Vi is colored j}                             

Which of the following statements is/are TRUE?

(A) This procedure results in a proper vertex coloring of G.
(B) The number of colors used is at most Δ(G)+1.
(C) The number of colors used is at most Δ(G).
(D) The number of colors used is equal to the chromatic number of G.

Answer: A and B