Download MC302 GRAPH THEORY Thursday, 11/21/13 (revised slides, 11/25

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