Packing Degree Sequences
Packing Degree Sequences
Tyler Seacrest
Joint work with Stephen Hartke
January 13, 2010
Packing Degree Sequences
Table of contents
1 Example
2 Packing Graphs
3 Degree Sequences
4 Packing Degree Sequences
5 Necessary and Sufficient?
Packing Degree Sequences
Example
A Party
3 Friends
1 Acquaintance
2 Friends
0 Acquaintances
2 Friends
2 Acquaintances
1 Friend
2 Acquaintances
0 Friends
2 Acquaintances
1 Friend
3 Acquaintances
Packing Degree Sequences
Packing Graphs
Packing Graphs
Two graphs G and H pack if they can be drawn on the same n
vertices without overlapping edges.
Packing Degree Sequences
Packing Graphs
Packing Graphs
Two graphs G and H pack if they can be drawn on the same n
vertices without overlapping edges.
Packing Degree Sequences
Packing Graphs
Packing Graphs
Two graphs G and H pack if they can be drawn on the same n
vertices without overlapping edges.
The above two graphs pack as shown, but they do not naively
pack. That is, some permutation of the second graph is needed
before they pack.
Packing Degree Sequences
Packing Graphs
Packing Graphs
Let ∆(G ) denote the maximum degree of G . The big conjecture in
graph packing:
Packing Degree Sequences
Packing Graphs
Packing Graphs
Let ∆(G ) denote the maximum degree of G . The big conjecture in
graph packing:
Conjecture (Bollobás-Eldridge)
(∆(G ) + 1)(∆(H) + 1) ≤ n + 1 implies that G and H pack.
Packing Degree Sequences
Degree Sequences
Degree Sequences
Let G be a graph. If you take the degree of each vertex and put
them into a sequence π, what you have is a graphic degree
sequence. We say that G realizes π.
Packing Degree Sequences
Degree Sequences
Degree Sequences
Let G be a graph. If you take the degree of each vertex and put
them into a sequence π, what you have is a graphic degree
sequence. We say that G realizes π.
Example
π = (4, 3, 2, 2, 1),
Packing Degree Sequences
Degree Sequences
Degree Sequences
Let G be a graph. If you take the degree of each vertex and put
them into a sequence π, what you have is a graphic degree
sequence. We say that G realizes π.
Example
π = (4, 3, 2, 2, 1),
3
G=
4
2
1
2
Packing Degree Sequences
Packing Degree Sequences
Packing Degree Sequences
Let α and β be two degree sequences. There are two notions of
how to talk about packing degree sequences:
Packing Degree Sequences
Packing Degree Sequences
Packing Degree Sequences
Let α and β be two degree sequences. There are two notions of
how to talk about packing degree sequences:
1
α and β pack if there exist graph realizations A and B that
pack. In this case, you may permute which coordinate of α
goes with which coordinate of β.
Packing Degree Sequences
Packing Degree Sequences
Packing Degree Sequences
Let α and β be two degree sequences. There are two notions of
how to talk about packing degree sequences:
1
α and β pack if there exist graph realizations A and B that
pack. In this case, you may permute which coordinate of α
goes with which coordinate of β.
2
α and β pack if there exist graph realizations A and B that
naively pack. In this case, you may not permute which
coordinate of α goes with which coordinate of β.
Packing Degree Sequences
Packing Degree Sequences
Packing Degree Sequences
Let α and β be two degree sequences. There are two notions of
how to talk about packing degree sequences:
1
α and β pack if there exist graph realizations A and B that
pack. In this case, you may permute which coordinate of α
goes with which coordinate of β.
2
α and β pack if there exist graph realizations A and B that
naively pack. In this case, you may not permute which
coordinate of α goes with which coordinate of β.
Packing Degree Sequences
Packing Degree Sequences
Packing Degree Sequences
Example
α = (3, 2, 2, 1, 0), β = (1, 0, 2, 3, 2).
Packing Degree Sequences
Packing Degree Sequences
Packing Degree Sequences
Example
α = (3, 2, 2, 1, 0), β = (1, 0, 2, 3, 2).
1 If we permute β so that it is equal to (0, 1, 2, 2, 3), then α and
β pack:
2, 1
3, 0
0, 3
2, 2
1, 2
Packing Degree Sequences
Packing Degree Sequences
Packing Degree Sequences
Example
α = (3, 2, 2, 1, 0), β = (1, 0, 2, 3, 2).
2 If we cannot permute β, we are forced to have an overlapping
edge, and thus α and β do not pack:
2, 0
3, 1
0, 2
2, 2
1, 3
Packing Degree Sequences
Packing Degree Sequences
Degree Sequence Packing
However, we may as well study only the second kind of degree
packing:
Packing Degree Sequences
Packing Degree Sequences
Degree Sequence Packing
However, we may as well study only the second kind of degree
packing:
Lemma
Let α = (a1 ≥ · · · ≥ an ) and β = (b1 ≥ · · · ≥ bn ). Then if α and
β pack given some permutation of β, they pack given the
permutation β = (bn , bn−1 , . . . , b1 ).
Packing Degree Sequences
Packing Degree Sequences
Degree Sequence Packing
However, we may as well study only the second kind of degree
packing:
Lemma
Let α = (a1 ≥ · · · ≥ an ) and β = (b1 ≥ · · · ≥ bn ). Then if α and
β pack given some permutation of β, they pack given the
permutation β = (bn , bn−1 , . . . , b1 ).
In other words, the permutation where the α-degrees are
non-increasing and the β-degrees are non-decreasing is the
easiest to pack.
Packing Degree Sequences
Packing Degree Sequences
Degree Sequence Packing
However, we may as well study only the second kind of degree
packing:
Lemma
Let α = (a1 ≥ · · · ≥ an ) and β = (b1 ≥ · · · ≥ bn ). Then if α and
β pack given some permutation of β, they pack given the
permutation β = (bn , bn−1 , . . . , b1 ).
In other words, the permutation where the α-degrees are
non-increasing and the β-degrees are non-decreasing is the
easiest to pack.
Thus, the first kind of degree packing is a special case of the
second kind.
Packing Degree Sequences
Necessary and Sufficient?
Searching for a Necessary and Sufficient Condition
Again let α = (a1 , . . . , an ), β = (b1 , . . . , bn ). Let
α + β = (a1 + b1 , . . . , an + bn ).
Packing Degree Sequences
Necessary and Sufficient?
Searching for a Necessary and Sufficient Condition
Again let α = (a1 , . . . , an ), β = (b1 , . . . , bn ). Let
α + β = (a1 + b1 , . . . , an + bn ).
Necessary condition for α and β to pack: α + β is a graphic
degree sequence.
Packing Degree Sequences
Necessary and Sufficient?
Searching for a Necessary and Sufficient Condition
Again let α = (a1 , . . . , an ), β = (b1 , . . . , bn ). Let
α + β = (a1 + b1 , . . . , an + bn ).
Necessary condition for α and β to pack: α + β is a graphic
degree sequence.
Not Sufficient
Packing Degree Sequences
Necessary and Sufficient?
Searching for a Necessary and Sufficient Condition
Again let α = (a1 , . . . , an ), β = (b1 , . . . , bn ). Let
α + β = (a1 + b1 , . . . , an + bn ).
Necessary condition for α and β to pack: α + β is a graphic
degree sequence.
Not Sufficient
Packing Degree Sequences
Necessary and Sufficient?
Searching for a Necessary and Sufficient Condition
Example of α + β being graphic not being sufficient
α = (3, 2, 2, 1, 0, 0), β = (1, 0, 0, 3, 2, 2), α + β = (4, 2, 2, 4, 2, 2)
Packing Degree Sequences
Necessary and Sufficient?
Searching for a Necessary and Sufficient Condition
Example of α + β being graphic not being sufficient
α = (3, 2, 2, 1, 0, 0), β = (1, 0, 0, 3, 2, 2), α + β = (4, 2, 2, 4, 2, 2)
α + β realization
Packing Degree Sequences
Necessary and Sufficient?
Searching for a Necessary and Sufficient Condition
Example of α + β being graphic not being sufficient
α = (3, 2, 2, 1, 0, 0), β = (1, 0, 0, 3, 2, 2), α + β = (4, 2, 2, 4, 2, 2)
α + β realization
2
2
4
4
2
2
Packing Degree Sequences
Necessary and Sufficient?
Searching for a Necessary and Sufficient Condition
Example of α + β being graphic not being sufficient
α = (3, 2, 2, 1, 0, 0), β = (1, 0, 0, 3, 2, 2), α + β = (4, 2, 2, 4, 2, 2)
α + β realization
2
α and β realizations
2
4
4
2
2
Packing Degree Sequences
Necessary and Sufficient?
Searching for a Necessary and Sufficient Condition
Example of α + β being graphic not being sufficient
α = (3, 2, 2, 1, 0, 0), β = (1, 0, 0, 3, 2, 2), α + β = (4, 2, 2, 4, 2, 2)
α + β realization
2
α and β realizations
2
2, 0
4
4
2
2
3, 1
0, 2
2, 0
1, 3
0, 2
Packing Degree Sequences
Necessary and Sufficient?
Searching for a Necessary and Sufficient Condition
The smallest number appearing in degree sequence π is
denoted δ(π).
Packing Degree Sequences
Necessary and Sufficient?
Searching for a Necessary and Sufficient Condition
The smallest number appearing in degree sequence π is
denoted δ(π).
Sufficient Condition:
Packing Degree Sequences
Necessary and Sufficient?
Searching for a Necessary and Sufficient Condition
The smallest number appearing in degree sequence π is
denoted δ(π).
Sufficient Condition:
Theorem
If α + β is graphic and
p
p
√
δ(α) + δ(β) ≥ n, then α and β pack.
Packing Degree Sequences
Necessary and Sufficient?
Searching for a Necessary and Sufficient Condition
The smallest number appearing in degree sequence π is
denoted δ(π).
Sufficient Condition:
Theorem
If α + β is graphic and
Not Necessary
p
p
√
δ(α) + δ(β) ≥ n, then α and β pack.
Packing Degree Sequences
Necessary and Sufficient?
Searching for a Necessary and Sufficient Condition
Question: Can we generalize Erdős-Gallai?
Theorem
Erdős-Gallai Let π = (d1 , . . . , dn ) be a degree sequence. Then π is
graphic if and only if
X
X
di ≤ |A|(|A| − 1) +
min{di , k}
i∈A
for all A ⊆ {1, . . . , n}.
i6∈A