 # Question: Is Every 4 Colourable Graph Planar?

## How do you prove a graph is not 3-colorable?

Larger classes of graphs A slightly more general result is true: if a planar graph has at most three triangles then it is 3-colorable.

However, the planar complete graph K4, and infinitely many other planar graphs containing K4, contain four triangles and are not 3-colorable..

## How do you know if a graph is not planar?

Theorem: [Kuratowski’s Theorem] A graph is non-planar if and only if it contains a subgraph homeomorphic to K_{3,3} or K_5. A graph is non-planar iff we can turn it into K_{3,3} or K_5 by: Removing edges and vertices.

## What are the main parts of the planar graph?

When a connected graph can be drawn without any edges crossing, it is called planar . When a planar graph is drawn in this way, it divides the plane into regions called faces . Draw, if possible, two different planar graphs with the same number of vertices, edges, and faces.

## How long did it take to prove the 4 Colour map theorem?

a thousand hoursA computer-assisted proof of the four color theorem was proposed by Kenneth Appel and Wolfgang Haken in 1976. Their proof reduced the infinitude of possible maps to 1,936 reducible configurations (later reduced to 1,476) which had to be checked one by one by computer and took over a thousand hours .

## Are all cycle graphs planar?

A Halin graph is a graph formed from an undirected plane tree (with no degree-two nodes) by connecting its leaves into a cycle, in the order given by the plane embedding of the tree. Equivalently, it is a polyhedral graph in which one face is adjacent to all the others. Every Halin graph is planar.

## How do you know if a graph is three colorable?

Let x be a vertex in V (G) − (N[v] ∪ N2(v)). In any proper 3-coloring of G, if it exists, the vertex x either gets the same color as v or x receives a different color than v. Therefore it is enough to determine if any of the graphs G/xv and G ∪ xv are 3-colorable. Recall that by our hypothesis d(x) ≥ 8.

## What is a K4 graph?

K4 is a maximal planar graph which can be seen easily. In fact, a planar graph G is a maximal planar graph if and only if each face is of length three in any planar embedding of G. Corollary 1.8. 2: The number of edges in a maximal planar graph is 3n-6.

## What is the chromatic number of each graph?

A graph coloring is an assignment of labels, called colors, to the vertices of a graph such that no two adjacent vertices share the same color. The chromatic number χ(G) of a graph G is the minimal number of colors for which such an assignment is possible.

## Is K2 a planar graph?

The graphs K2,2,2,2,1 and K2,2,2,2,2 are not 1-planar because they contain K5,4 as a subgraph.

## Is Grotzsch graph planar?

Theorem 4.6: Grötzsch’s Graph is non-planar.

## What is chromatic number in graph theory?

The chromatic number of a graph is the smallest number of colors needed to color the vertices of so that no two adjacent vertices share the same color (Skiena 1990, p. 210), i.e., the smallest value of. possible to obtain a k-coloring.

## Can a non planar graph be 4 colors?

3 Answers. Obviously not. A graph is bipartite if and only if it is 2-colorable, but not every bipartite graph is planar (K3,3 comes to mind).

## Is every planar graph 3-colorable?

Every planar graph without adjacent 3-cycles and without 5-cycles is 3-colorable. (By intersecting (adjacent) triangles we mean those with a vertex (an edge) in common.)

## Which of the following graphs is not planar?

Example: The graphs shown in fig are non planar graphs. These graphs cannot be drawn in a plane so that no edges cross hence they are non-planar graphs.

## What is the three color problem?

This issue is a part of graph theory. It is well known that, given any separation of a plane into contiguous regions, producing a figure called a map, no more than four colors are required to color the regions of the map so that no two adjacent regions have the same color.

## What is N colorable?

A graph is said to be k-colorable if it can be properly colored using k colors. … Conversely, if a graph is 2-colorable, then the vertices having same color can be taken as disjoint sets. Hence, we arrive at the following result: Theorem: A graph is bipartite if and only if it is 2-colorable.

## Which of the following graph is not 3-colorable?

Almost all graphs with 2.522 n edges are not 3-colorable. Electronic Journal of Combinatorics, 6(1), R29.