Graph Coloring, Planar Graph and Partial Order

1. Graph Coloring, Planar Graph and Partial Order CSC2110 Tutorial 5 Jerry Le jlle@cse.cuhk.edu.hk Rm120, SHB Office hour: Thu 3pm-5:30pm Oct 18 2007

2. Outline • Quick Review • Q & A (questions from classwork2)

3. Quick Review • Graph Coloring - Concept (Chromatic number, x-colorable...) - Applications(Map coloring, Server upgrading, Register allocation...) • Planar Graph - Concept (Planar embedding, Discrete faces) - Important Theorems (Euler’s formula, 6-color theorem) • Partial order - Definition (antichain, chain, directed graph) - Properties (Transitive, Asymmetric) - Important Theorems (Dilworth’s Theorem) - Applications (Parallel task scheduling, Monotone sequences...)

4. Q & A(questions from classwork 2)

5. True or False (Graph coloring) 1. A bipartite graph is 2-colorable. - True, put one color in one side, another color in the other 2. A 2-colorable graph is a bipartite graph. - True 3. A tree is 2-colorable. - True, because it has no cycles. 4. A 2-colorable graph is a tree. - False, it may not be connected, or may have cycles. 5. We have a k+1 vertex coloring for each graph with maximum degree k, and this coloring is optimal. - False, consider this

6. True or False (Planar graph) 1. Any connected graph has a planar embedding. - False, need to be planar graph... 2. Any graph of 5 vertices is a planar graph. - False, one counterexample is K5 3. Every simple planar graph is 6-colorable. - True. 4. The vertex degree of each simple planar graph is at most 5. - False, but we at least have one vertex of degree at most 5

7. True or False (Partial order) 1. “≤” can represent a partial order. - True, transitive and asymmetric 2. The subset relation “ ” can represent a partial order - True, check it out. 3. Let directed graph D represent a partial order, then D has no cycles. - True, otherwise the order is not asymmetric. 4. There are n elements in a partial order, and the length of the critical path (i.e. longest chain) is less than t, then there is an antichain of size at least n/t. - True, Dilworth’s theorem 5. Every partial ordering on a non-empty finite set has a minimal element. - True

8. Vertex coloring Let G be a graph. Which of the following statements are equivalent to each other? 1. G contains no odd-length cycles. 2. G is bipartite. 3. G is 2-colorable. 4. The maximum vertex degree of G is 1. 1,2,3 are equivalent.

9. Vertex coloring Find out the chromatic number of the following graphs.

10. Vertex coloring Find out the chromatic number of the following graphs.

11. Application of Vertex Coloring: Register Allocation (Long Question Q5.) When a computer performs calculations, the values of variables are stored in the memory. Register is a type memory that is very fast but also very expensive. Therefore, we want to use registers as efficiently as possible. Suppose we need to do the following calculation: Of course we can allocate a unique register for each variable, but that is not efficient enough... For example, b and d can use the same register, because we don’t need b anymore after the 1st step.

12. Application of Vertex Coloring: Register Allocation (Long Question Q5.) Suppose we need to do the following calculation: Of course we can allocate a unique register for each variable, but that is not efficient enough... For example, b and d can use the same register, because we don’t need b anymore after the 1st step. But we do need to let some variables use separate register, for example, a and b in the first step, c and d, etc.

13. Application of Vertex Coloring: Register Allocation (Long Question Q5.) Suppose we need to do the following calculation: a) Try to construct a graph to help you decide to use how many registers in total. Basically you need to decide: what do the variables represent? Under what condition should there be an edge between two vertices.

14. Application of Vertex Coloring: Register Allocation (Long Question Q5.) Suppose we need to do the following calculation: Generally, we operate each variable in two ways: “use” and “assignment” For example, c=a+b, we are “using” a and b, and we are “assigning” c an value. We call the interval between the first assignment (or input) and last use (or output) of a variable as its “lifetime” Two variables can not be assigned the same register if their lifetimes overlap!

15. Application of Vertex Coloring: Register Allocation (Long Question Q5.) Suppose we need to do the following calculation: Each vertex represents a variable, and we add an edge between two vertices iff their lifetimes overlap.

16. Application of Vertex Coloring: Register Allocation (Long Question Q5.) Suppose we need to do the following calculation: Find a vertex coloring of this graph, we need at least 4 colors (registers).

17. Vertex Coloring and Planar Graph (Long Question Q6.) 1. Prove that every triangle-free connected planar graph has a vertex of degree at most 3. 2. Prove that every triangle-free connected planar graph is 4-colorable. For triangle-free graph, every face is bounded by at least 4 edges, we have 4f ≤ 2e Using Euler’s theoremv-e+f=2, we have e ≤ 2v-4 If all vertices are of degree at least 4, then we have 4v ≤ 2e e ≤ e-4, contradiction! We proved the first statement.

18. Vertex Coloring and Planar Graph (Long Question Q6.) 1. Prove that every triangle-free connected planar graph has a vertex of degree at most 3. 2. Prove that every triangle-free connected planar graph is 4-colorable. Using result from the first statement 1, we can easily prove the second one, by adopting the same technique of proving “6-color theorem” (check it in the lecture notes).

19. Application of Dilworth’s theorem (Long Question Q7.) We are given a real number sequence Prove that the sequence must contain a monotone subsequence of length n+1. Define (i, ai) as the element in the partial order We say (i, ai) ≤ (j, aj) if i ≤ j and ai ≤ aj Now we have a partial order on the set {(i, ai)}

20. Application of Dilworth’s theorem (Long Question Q7.) We are given a real number sequence Prove that the sequence must contain a monotone subsequence of length n+1. A chain in our partial order is a monotone increasing subsequence An antichain in our partial order is a monotone decreasing subsequence Using Dilworth’s theorem, we are done!