Introduction to Graph Theory Lecture 13: Graph Coloring: Edge Coloring

The Edge-Chromatic Number Proper edge coloring: incident edges receive distinct colors. The edge-chromatic number, : the minimum number of colors required for proper edge coloring What is the lower bound for ?

Edge Coloring Technique To color a complete bipartite graph, we need only The technique is to rotate the colors

Complete Graph Theorem 6.6: For the complete graph, we have Proof: (using the rotation technique)  Graph has edges.  A maximum matching contains n-1 edges, and we can assign one color to these edges.  Therefore, we need at least 2n-1 colors for proper edge coloring.  How can we achieve that?

(cont) We do the max matching in the following order: 2n-1 2n-2 n+1 n n-1 n+2 n … 4 …

(cont) Continue rotating, and when j is the unmatched vertex, color the matched edges. This defines a proper (2n-1)-edge coloring of

Complete Graph Theorem 6.7: For the complete graph, we have Proof:  The same technique as for  But match the unmatched vertex i with a new vertex labeled 2n  In other words, we have perfect matching for each i Generalization: For any graph G,

for Bipartite Graph Theorem 6.9: If G is a bipartite graph, then Proof: The proof is divided into two parts  Part1: show that it is true for -regular bipartite graph.  Part2: show that if G is a bipartite graph of, then G can always be embedded in a -regular bipartite graph.

(Proof: Part1) Proof by induction  Basic case when, we have n copies of K 2, so  Inductive Hypothesis (IH): Assume true when  Let G be a -regular bipartite graph  Using theorem 3.7 we know that G has a perfect matching M. Color those edges with color and remove them from G.  From IH, we know that  Together with M,, but

(Proof: Part2) Now convert G to a -regular bipartite graph  By lemma 3.7: -regular bipartite graph must be equitable.  Adding vertices and edges if necessary to make G a -regular graph H.  From Part1 we know that   But is lower-bounded by. 

Monochromatic Triangle in This is a more relaxed setting: no implication that edges that share a vertex have distinct color. monochromatic bichromatic Ambivalent vertices

(cont) Q1: How many bichromatic triangles can we get in, given the function r(i) which is the number of red edges incident with vertex i? Q2: How many monochromatic triangles can we get in ?

Theorem 6.10 Theorem: Given an edge coloring of using two colors, day red and blue, and given the function, which yields the number of red edges incident with vertex i, then the number of monochromatic triangles is given by

Proof Count the bichromatic triangles first  If vertex i is ambivalent, then a triangle is formed by selecting a red and an blue edge incident with vertex i.  There are and ways of choosing the red edge and blue edge, respectively.  That gives us bichromatic triangles in which vertex i is ambivalent.  But each bichromatic triangle has two ambivalent vertices. So each bichromatic triangle is counted twice.  There are number of triangles in, so # of monochromatic triangle can be obtained by subtracting # of bichromatic triangles.

Lower Bound Can we color, such that we have no monochromatic triangle? Can this be done for any with ?  NO! We’ll see the formula for the lower bound for the number of monochromatic triangles in when and.

Lemma 6.11 Let’s prove the lemma first Let m be a given positive number. Then among all pairs of positive numbers x and y, such that, the product is maximum when Proof:  Follow the argument in the textbook, or  Find the maximum of

Corollary When m, x, and y are positive integers where x+y=m, we maximize xy when and

The Formula for Lower Bond We want to minimize the expression derived in Theorem 6.10 Since the 1 st term is fixed, so we can maximize the 2 nd term (i.e. maximizing each term of the summation). 1 st term 2 nd term

(cont) Using the lemma and corollary, is maximized when and This gives us

Applications of Graph Coloring Designing a modern zoo that allows as much exercise room as possible for each animal, but enclosure is needed to separate a predator from its prey. How to minimize the number of enclosures?

Applications of Graph Coloring Example 6.7 for exam scheduling. are teachers, and are classes. meets with class a total of times per week. We want to work out the minimum number of periods per week in the timetable. t3 t2 t1 t4 t5 c1c3c4c5c6c7c