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Graph Theory: Planar Graphs

Continuing from the previous post ( which is here ). This is a bit short post. I think this post is not at all helpful in olys but then let's do it cause I find them interesting :P. Here are my notes..

Book referred: Daniel A Marcus Graph theory

Chapter H:

  • Planar Graphs: A planar graph is a graph that can be represented by a diagram with no edge cross. Such a diagram is called a plane diagram. For K_4 is a planar graph.
  • Regions formed by a graph: The planar graph divides the plan into regions 
  • Non- planar graph: A non-planar graph is a graph that always has a edge cross. 
  • Regional Degrees: The regions are boundaries by edges. The number od edges each region is boundaries with is called regional degrees.
  • Subdivision: A graph that is obtained by inserting vertices of degree 2 is an already existing edge.



Regional Degree theorem: Let G be a connected graph and let r_1,r_2,\dots be the degrees of the regions in any plane diagram of G. Then the sum of r_1+r_2+r_3+\dots =2e
    Proof: Note that each edges contributes two times in the region sum, since it's part of two regions. ( True for multi graph too)

      Funfact: Let G be a connected graph with three or more vertices. Then in any plane of diagrams of G, every region has degree greater than or equal to 3.
        Regions and each edge contribute to two

          Euler's Formula: Let G be a connected planar graph with v vertices and e edges and r be the number of regions in any plane diagram G. Then v-e+r=2.
            More on Euler's formula: We can extend it to solid to reknown formula v-e+f=2.
              Proof: We consider a spanning tree of the  connected graph. It exist as the graph is connected. Note that now we have v-e+r= v-(v-1)+1=2. 
                Now, whenever we add an edge, we get a new cycle and hence a new region. So the -e+r is constant.
                  Another proof: Reduce the graph into spanning tree by  removing one edge from a cycle and same reasoning. 
                    Problem 1: Prove that K_5 is non planar.
                      Proof: Suppose it is planar, then since K_5 is connected, it must satisfy the Euler's formula. So v-e+r=2\implies r= 7
                        But by the funfact, the regional degree sums must be \ge 7\cdot 3=21 but regional degree sum is 2e=20. Contradiction.
                          Problem 2: Let G be a connected graph with three or more vertices. Show that r\le 2e/3 and prove that e\le 3v-6.
                            Proof: For the first inequality we use the fact that each region is bondried by atleats 3 edges. And 2e=\sum r_i, r_i\ge 3 \implies r\le 2e/3. Using euler's formula, we get v-e+r=2\implies v-e+2e/3\ge 2\implies 3v-6 \ge e
                              Problem 3: In any plane diagram of a bipartite graph, all of the regional degrees must be even. 
                                Proof: Consider the boundaries of any region. Since it is a bipartite graph, colour the vertices with two colour such that no two adjacent vertices are of the same colour. So the boundaries must have even length i.e even degree.
                                  Problem 4: Suppose G is bipartite Show that r\le e/2 and e\le 2v-4.
                                    Prooof: Since G is bipartite, the regional degree must be at least 4 so r\le \frac{2e}{4}=\frac{e}{2}.
                                      Using Euler's Formula, we get v-e+r=2\implies v-e +\frac{e}{2}\ge 2\implies 2v-e\ge 4
                                        So e\le 2v-4.
                                          Problem 5: Prove that K_{3,3} is non planar.
                                            Proof: Suppose it is planar, then since K_{3,3} is connected, it must satisfy the Euler's formula.
                                              By euler's formula, we get v-e+r=2\implies r= 5. Since, we have bipartite graph, we get r\le e/2 but 5\ge 9/2.
                                                Non planar graphs [intuitive idea]: 
                                                  When we add a degree two vertex to a  non plannar graph G, the new subdivision graph G' would be non plannar. In order to make G a planar graph, we would have to add vertex in the "intersection" of the edges, which have degree 4.
                                                    Moreover, if the subdivision G' of the graph G is planar then G is planar too. The only case it wont be planar is when we remove the vertex from G' which forms the intersection point. But then this vertex would have degree 4 at least. Hence it is not the point we added so form the subdivision.
                                                      Hence, if one graph is a subdivision of another, then either both graphs are planar or else both are non planar.
                                                      Subdivision of K_5

                                                        Problem: If a graph G contains a subgraph with is either K_{3,3} or K_5 or a subdivision of one of these, then G must be non -planar.
                                                          This just works, because we know the subdivision would be non planar.

                                                              Now, here's the black magic! I don't know the proof of this theorem though.. 

                                                                  Kuratowski's Theorem: Every non planar contains a subgraph which is either K_5 or K_{3,3}.
                                                                    Using this theorem, we can show that a Petersen graph non planar too.


                                                                    The Petersen graph is non planar

                                                                    Problem: Show that the 
                                                                    Petersen graph is non-planar 

                                                                    Proof: Refer to the below image 


                                                                    We use the subdivision fact and the fact that K_{3,3} is non-planar. By Kuratowski's Theorem we are done.
                                                                      Yayy!! That's it for this blog post, I think there will be a sequel talking about vertex colouring which I am so excited to learn! 
                                                                        Sunaina💜

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