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MC302 GRAPH THEORY Thursday, 11/21/13 (revised slides, 11/25/13) Today: Clique number vs. Chromatic Number Edge coloring Reading: [CH] 6.5 [HR] 2.2 Exercises: [CH] p. 218: 6.5.3 [HR] p. 35: 2.2.4, 2.2.5, 2.2.7 Thursday, 11/21/13, Slide #1 Chromatic Number vs. Clique Number We know that if is a subgraph of , then . But we’ve also seen that they do not have to be equal. Definition: The clique number of is the largest such that is a subgraph of . Thus . A perfect graph G is one with the property that, for every induced subgraph H of G (including G itself), . Thursday, 11/21/13, Slide #2 Edge-coloring and An edge-coloring of is an assignment of colors to its edges so that adjacent edges have different colors A -edge coloring is one that uses colors. The edge-chromatic number is the smallest such that G is -colorable. Also denoted ′. Thursday, 11/21/13, Slide #3 Edge-chromatic number Maximum degree vs. For vertex chromatic number, Brooks’ Theorem: ∆ unless is a complete graph or odd cycle, in which case ∆ + 1. can be much less than ∆: , , but , . For edge-chromatic number, it’s clear that ∆ . But how far apart can they be? Vizing’s Theorem. Δ Δ + 1. Thursday, 11/21/13, Slide #4 Edge-chromatic number and line graphs (revised) The edge-chromatic number of equals the vertex-chromatic number of : For G connected: Except when , ∆ Vizing’s Theorem says, for all graphs : -clique in L(G) ↔ degree- vertex in G This implies that, except for , ∆ or ∆ + Together, this says, for all line graphs : or + I.e., line graphs are “almost perfect.” Thursday, 11/21/13, Slide #5 Snarks “Most” 3-regular graphs have 3. The 2-connected, 3-regular graphs that have 4 are called snarks. The Petersen Graph is a snark. Here are more, from mathworld.wolfram.com/Snark.html Thursday, 11/21/13, Slide #6 Edge-chromatic numbers of Bipartite Graphs (revised + new slides) Theorem. (König, 1916). If is a bipartite graph, then Δ. Lemma 1. If is bipartite, then is a subgraph of a Δ-regular bipartite graph H. Proof. First add vertices, if needed, to make both partite sets the same size. Then add edges to make all vertices have degree Δ. See next two slides! Lemma 2. If G is a regular bipartite graph, then G has a perfect matching. (already proved) Proof of Theorem. Use Lemma 1, and then use Lemma 2, Δ times. Thursday, 11/21/13, Slide #7 A counterexample to the construction of Lemma 1 Take any regular bipartite graph with # 3: Replace any one edge as follows: The resulting graph cannot be made ∆-regular by just adding edges! But there’s another way! Thursday, 11/21/13, Slide #8 A Construction that works for any graph (bipartite or not) Proposition. If is any graph, then there is a regular graph $ with the property that is a subgraph of $ and ∆ $ ∆ . Proof. If G has any vertices with deg ) * ∆, make a new graph ′, by taking two copies of and adding an edge between any such vertex v in and its copy )’ in ’. If G’ is not regular, repeat process until a regular graph is obtained. Thursday, 11/21/13, Slide #9 Application: Latin Squares A Latin Square of order n is an , - , matrix with the numbers 1,2, … , ,in each row and column, with no repeated number in any row or column These correspond to edgecolorings of bipartite graphs: If 0,0 has partition 1 ∪ 3, and edge 45 67 has color , then put row 8, 9. [CH] does this slightly differently. y1 y2 y3 x1 2 3 1 x2 1 2 3 x3 3 1 2 1=blue 2=green 3-red Thursday, 11/21/13, Slide #10 Edge-chromatic number of the complete graph 0 Theorem. If , 3 is odd, then 0 , Δ 0 + 1. If , 2 is even, then 0 , − 1 Δ0 . Lemma 1. For , odd, 0 > , − 1 Δ 0 . Proof by contradiction. No color can be used 0< more than times … = Lemma 2. For , odd, 0 has an edge-coloring with , colors. Proof on next slide. Lemma 3. For n even, an , − 1-edge coloring of 0< extends to an , − 1-edge coloring of 0 . Proof on next slide. Thursday, 11/21/13, Slide #11 Finishing proof for 0 For , odd, Color outer edges 1 to ,. Color inner edges same color as the parallel outer edge. At each vertex, no edge uses the color of the opposite edge. For , even, Remove one vertex and edge-color 0< . Add the ,>? vertex adjacent to each other vertex using missing color on its edge. Thursday, 11/21/13, Slide #12 Application: Scheduling Games Suppose we have an even number , of teams, and each week each team plays some other team. Can we schedule games each week so that every team plays every other team exactly once? Answer: Each week corresponds to an edgecoloring of 0 . Thursday, 11/21/13, Slide #13