Thursday, 11/21/13, Slide #1

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 #2

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 #3

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 #4

Edge-chromatic number � vs.

Maximum degree

� 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 #5

Edge-chromatic number and line graphs

(revised)

� The edge-chromatic number of � equals the

vertex-chromatic number of ����:

� �� � � � � �

� For G connected: Except when � � ��,

� -clique in L(G) ↔ degree- vertex in G

� This implies that, except for ��,

� � � � � ∆���

� Vizing’s Theorem says, for all graphs �:

� �� � � ∆��� or ∆ � + �

� Together, this says, for all line graphs ����:

� � � � � � � � or � � � + �

� I.e., line graphs are “almost perfect.”

Thursday, 11/21/13, Slide #6

Snarks

� “Most” 3-regular graphs have �� � 3.

� The 2-connected, 3-regular graphs that have �� � 4are called snarks.

� The Petersen Graph is a snark.

� Here are more, from mathworld.wolfram.com/Snark.html

Thursday, 11/21/13, Slide #7

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 ���-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 ���.

� 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, ��� times.

Thursday, 11/21/13, Slide #8

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 #9

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 #10

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 edge-colorings of bipartite graphs: If �0,0 has

partition 1 ∪ 3, and edge 4567 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 #11

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 � Δ��0�.

� Lemma 1. For , odd, �� �0 > , − 1 � Δ �0 .

� Proof by contradiction. No color can be used

more than 0<�

=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 #12

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 #13

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 edge-coloring of �0.