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.