Exploring Graph Coloring Techniques: Minimizing Colors for Proper Edge Coloring

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: 1 1 2n-1 2n-1 2 2 2n-2 n+2 … 3 3 … 4 n+1 n+1 n

(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 K2, 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. Ambivalent vertices monochromatic bichromatic

(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 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 1st term is fixed, so we can maximize the 2nd term (i.e. maximizing each term of the summation). 1st term 2nd 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 t4 t2 t5 t1 3 2 2 3 1 2 2 2 3 3 2 4 3 2 2 c1 c2 c3 c4 c5 c6 c7

Exploring Graph Coloring Techniques: Minimizing Colors for Proper Edge Coloring