-
Example – Is the hypercube
planar? -
Solution – Yes,
is planar. Its planar representation-
Regions in Planar Graphs –
The planar representation of a graph splits the plane into regions. These regions are bounded by the edges except for one region that is unbounded. For example, consider the following graph ”- Example – What is the number of regions in a connected planar simple graph with 20 vertices each with a degree of 3?
-
Solution – Sum of degrees of edges = 20 * 3 = 60. By handshaking theorem,
which gives . By Euler’s theorem, the number of regions = which gives 12 regions.
-
Example – Is the graph
planar? -
Solution – Number of vertices and edges in
is 5 and 10 respectively. Since 10 > 3*5 – 6, 10 > 9 the inequality is not satisfied. Thus the graph is not planar.
Graph Coloring –
If you ever decide to create a map and need to color the parts of it optimally, feel lucky because graph theory is by your side. What is the maximum number of colors required to color the regions of a map? This question along with other similar ones have generated a lot of results in graph theory. First, let us define the constraint of coloring in a formal way- Coloring – “A coloring of a simple graph is the assignment of a color to each vertex of the graph such that no two adjacent vertices are assigned the same color.” A simple solution to this problem is to color every vertex with a different color to get a total of-
Example 1 – What is the chromatic number of the following graphs?
-
Solution – In graph
, the chromatic number is atleast three since the vertices , , and are connected to each other. The following color assignment satisfies the coloring constraint – – Red – Green – Blue – Red – Green – Blue – Red Therefore the chromatic number of is 3. In graph since and are also connected, therefore the chromatic number if 4. -
Example 2 – What is the chromatic number of
? -
Solution – Since every vertex is connected to every other vertex in a complete graph, the chromatic number is
. -
Example 3 – What is the chromatic number of
? -
Solution – If the vertex are colored in an alternating fashion, the cycle graph requires 2 colors. If
is odd, then the last vertex would have the same color as the first vertex, so the chromatic number will be 3. But if it is even, then first and last vertices will be of different color and the chromatic number will be 2. -
Example 4 – What is the chromatic number of
? -
Solution – In the bi-partite graph
, the vertices in are divided into two sets, such that there is no edge between vertices in the same set. Therefore the chromatic number of any bipartite graph is 2. One set of vertices can be assigned one color and the other can be assigned a different color for a total of 2 colors. It will satisfy the coloring constraint since the vertices of the same set are not connected.
GATE CS Corner Questions
Practicing the following questions will help you test your knowledge. All questions have been asked in GATE in previous years or in GATE Mock Tests. It is highly recommended that you practice them. 1. GATE CS 2012, Question 21 2. GATE CS 2011, Question 17 3. GATE CS 2009, Question 2 4. GATE CS 2008, Question 23 5. GATE CS 2005 Question 10 6. GATE CS 2005, Question 47 7. GATE CS 2004, Question 77 8. GATE CS 2002, Question 4 9. GATE CS 2015 Set-1, Question 63 10. GATE CS 2008, Question 3 11. GATE CS 2016 Set-2, Question 13References-
Planar Graphs – Wikipedia
Graph Coloring – Wikipedia
Discrete Mathematics and its Applications, by Kenneth H Rosen