Background
Perfect matchings with sparse intersection
Non-intersecting perfect matchings
in cubic graphs
Tomáš Kaiser
Department of Mathematics
and Institute for Theoretical Computer Science
University of West Bohemia
Pilsen, Czech Republic
Joint work with André Raspaud
Tomáš Kaiser
Non-intersecting perfect matchings in cubic graphs
Background
Perfect matchings with sparse intersection
Talk outline
1
Background
Perfect matchings in cubic graphs
Fano colorings and joins
Unions of perfect matchings
2
Perfect matchings with sparse intersection
A result for graphs of oddness two
Embedded graphs and balanced joins
Proof method
Problems
Tomáš Kaiser
Non-intersecting perfect matchings in cubic graphs
Background
Perfect matchings with sparse intersection
Perfect matchings in cubic graphs
Fano colorings and joins
Unions of perfect matchings
Perfect matchings in cubic graphs
Theorem (Petersen, 1891)
Every bridgeless cubic graph contains a perfect matching.
two disjoint perfect matchings need not exist:
Tomáš Kaiser
Non-intersecting perfect matchings in cubic graphs
Background
Perfect matchings with sparse intersection
Perfect matchings in cubic graphs
Fano colorings and joins
Unions of perfect matchings
The Berge–Fulkerson conjecture
Conjecture (Fulkerson, 1971; Berge)
Every bridgeless cubic graph has a double cover by six perfect
matchings.
equivalently: if G is cubic bridgeless, then the 6-regular graph
2G is 6-edge-colorable
any 3 of the perfect matchings would have empty intersection
Tomáš Kaiser
Non-intersecting perfect matchings in cubic graphs
Background
Perfect matchings with sparse intersection
Perfect matchings in cubic graphs
Fano colorings and joins
Unions of perfect matchings
The Fan–Raspaud conjecture
Conjecture (Fan and Raspaud, 1994)
Every bridgeless cubic graph contains three perfect matchings M1 ,
M2 and M3 such that
M1 ∩ M2 ∩ M3 = ∅.
open even if ‘3’ is replaced by any larger number (problem
posed by M. DeVos)
Tomáš Kaiser
Non-intersecting perfect matchings in cubic graphs
Background
Perfect matchings with sparse intersection
Perfect matchings in cubic graphs
Fano colorings and joins
Unions of perfect matchings
The Fan–Raspaud conjecture
Conjecture (Fan and Raspaud, 1994)
Every bridgeless cubic graph contains three perfect matchings M1 ,
M2 and M3 such that
M1 ∩ M2 ∩ M3 = ∅.
open even if ‘3’ is replaced by any larger number (problem
posed by M. DeVos)
Tomáš Kaiser
Non-intersecting perfect matchings in cubic graphs
Background
Perfect matchings with sparse intersection
Perfect matchings in cubic graphs
Fano colorings and joins
Unions of perfect matchings
Fano colorings
A Fano coloring of a cubic graph:
mapping of edges to points of F ,
3 edges incident with each point map to 3 points on a line of
F
001
101
011
011
101 110
111
110
100
110
010
011
101
Tomáš Kaiser
Non-intersecting perfect matchings in cubic graphs
Background
Perfect matchings with sparse intersection
Perfect matchings in cubic graphs
Fano colorings and joins
Unions of perfect matchings
Fano colorings (cont’d)
every 3-edge-colorable cubic graph has a Fano coloring
in fact, every bridgeless cubic graph has a Fano coloring
(Holroyd and Škoviera, 2004)
Tomáš Kaiser
Non-intersecting perfect matchings in cubic graphs
Background
Perfect matchings with sparse intersection
Perfect matchings in cubic graphs
Fano colorings and joins
Unions of perfect matchings
Joins
Definition
J ⊂ E (G ) is a join in G if each vertex v has the same degree in
(V (G ), J) and in G modulo 2
J is a join ⇐⇒ E (G ) − J has all degrees even,
G cubic =⇒ each perfect matching is a join.
Tomáš Kaiser
Non-intersecting perfect matchings in cubic graphs
Background
Perfect matchings with sparse intersection
Perfect matchings in cubic graphs
Fano colorings and joins
Unions of perfect matchings
Joins
Definition
J ⊂ E (G ) is a join in G if each vertex v has the same degree in
(V (G ), J) and in G modulo 2
J is a join ⇐⇒ E (G ) − J has all degrees even,
G cubic =⇒ each perfect matching is a join.
Tomáš Kaiser
Non-intersecting perfect matchings in cubic graphs
Background
Perfect matchings with sparse intersection
Perfect matchings in cubic graphs
Fano colorings and joins
Unions of perfect matchings
Joins and Fano colorings
let c be a Fano coloring of G
set Fi = {e : i-th coordinate of c(e) is 0}
011
101 110
110
011
011
100 110
101
101
001
010
each Fi is a join
some lines cause degree 3 vertices (e.g., 001, 100, 101 in F2 )
Tomáš Kaiser
Non-intersecting perfect matchings in cubic graphs
Background
Perfect matchings with sparse intersection
Perfect matchings in cubic graphs
Fano colorings and joins
Unions of perfect matchings
Joins and Fano colorings
let c be a Fano coloring of G
set Fi = {e : i-th coordinate of c(e) is 0}
011
101 110
110
011
011
100 110
101
101
001
010
each Fi is a join
some lines cause degree 3 vertices (e.g., 001, 100, 101 in F2 )
Tomáš Kaiser
Non-intersecting perfect matchings in cubic graphs
Background
Perfect matchings with sparse intersection
Perfect matchings in cubic graphs
Fano colorings and joins
Unions of perfect matchings
Joins and Fano colorings (cont’d)
green lines correspond to degree 1 vertices
in all Fi
each black line accounts for degree 3 in one
Fi
Observation
If we only use the green lines, the Fi will be 3 perfect matchings
with empty intersection.
similarly, the green+orange lines
correspond to a perfect matching and 2
joins with empty intersection
Tomáš Kaiser
Non-intersecting perfect matchings in cubic graphs
Background
Perfect matchings with sparse intersection
Perfect matchings in cubic graphs
Fano colorings and joins
Unions of perfect matchings
Joins and Fano colorings (cont’d)
green lines correspond to degree 1 vertices
in all Fi
each black line accounts for degree 3 in one
Fi
Observation
If we only use the green lines, the Fi will be 3 perfect matchings
with empty intersection.
similarly, the green+orange lines
correspond to a perfect matching and 2
joins with empty intersection
Tomáš Kaiser
Non-intersecting perfect matchings in cubic graphs
Background
Perfect matchings with sparse intersection
Perfect matchings in cubic graphs
Fano colorings and joins
Unions of perfect matchings
Joins and Fano colorings (cont’d)
green lines correspond to degree 1 vertices
in all Fi
each black line accounts for degree 3 in one
Fi
Observation
If we only use the green lines, the Fi will be 3 perfect matchings
with empty intersection.
similarly, the green+orange lines
correspond to a perfect matching and 2
joins with empty intersection
Tomáš Kaiser
Non-intersecting perfect matchings in cubic graphs
Background
Perfect matchings with sparse intersection
Perfect matchings in cubic graphs
Fano colorings and joins
Unions of perfect matchings
k-line Fano colorings
Definition
k-line Fano coloring: one that uses only ≤ k lines of F
As observed, e.g., by Máčajová and Škoviera,
3-colorable ⇐⇒ 1-line coloring
3 PMs with empty intersection ⇐⇒ 4-line coloring
2 PMs+join with empty intersection ⇐⇒ 5-line coloring
1 PM+2 joins with empty intersection ⇐⇒ 6-line coloring
3 joins with empty intersection ⇐⇒ 7-line coloring
Theorem (Máčajová and Škoviera, 2005)
Every bridgeless cubic graph has a 6-line Fano coloring.
they also noted that the existence of a 4-line coloring is
Tomáš Kaiser
Non-intersecting perfect matchings in cubic graphs
Background
Perfect matchings with sparse intersection
Perfect matchings in cubic graphs
Fano colorings and joins
Unions of perfect matchings
k-line Fano colorings
Definition
k-line Fano coloring: one that uses only ≤ k lines of F
As observed, e.g., by Máčajová and Škoviera,
3-colorable ⇐⇒ 1-line coloring
3 PMs with empty intersection ⇐⇒ 4-line coloring
2 PMs+join with empty intersection ⇐⇒ 5-line coloring
1 PM+2 joins with empty intersection ⇐⇒ 6-line coloring
3 joins with empty intersection ⇐⇒ 7-line coloring
Theorem (Máčajová and Škoviera, 2005)
Every bridgeless cubic graph has a 6-line Fano coloring.
they also noted that the existence of a 4-line coloring is
Tomáš Kaiser
Non-intersecting perfect matchings in cubic graphs
Background
Perfect matchings with sparse intersection
Perfect matchings in cubic graphs
Fano colorings and joins
Unions of perfect matchings
Joins and odd cuts
Observation
A set Y ⊂ E (G ) contains a join of G if and only if G − Y contains
no odd edge-cut of G .
Definition
A set X ⊂ E (G ) is sparse if X contains no odd edge-cut.
Corollary
X ⊂ E (G ) is sparse if and only if there is a join disjoint from X .
so finding a 5-line Fano coloring amounts to finding two
perfect matchings whose intersection is sparse
Tomáš Kaiser
Non-intersecting perfect matchings in cubic graphs
Background
Perfect matchings with sparse intersection
Perfect matchings in cubic graphs
Fano colorings and joins
Unions of perfect matchings
Joins and odd cuts
Observation
A set Y ⊂ E (G ) contains a join of G if and only if G − Y contains
no odd edge-cut of G .
Definition
A set X ⊂ E (G ) is sparse if X contains no odd edge-cut.
Corollary
X ⊂ E (G ) is sparse if and only if there is a join disjoint from X .
so finding a 5-line Fano coloring amounts to finding two
perfect matchings whose intersection is sparse
Tomáš Kaiser
Non-intersecting perfect matchings in cubic graphs
Background
Perfect matchings with sparse intersection
Perfect matchings in cubic graphs
Fano colorings and joins
Unions of perfect matchings
Unions of perfect matchings
how many edges of a bridgeless cubic graph G can be covered
by k perfect matchings?
define mk (G ) as
S
| Mi |
mk (G ) = max
.
M1 ,...,Mk |E (G )|
and set mk to be the infimum of mk (G ) over all bridgeless
cubic G
observe m1 = 1/3
the Berge–Fulkerson conjecture implies that m5 = 1
Tomáš Kaiser
Non-intersecting perfect matchings in cubic graphs
Background
Perfect matchings with sparse intersection
Perfect matchings in cubic graphs
Fano colorings and joins
Unions of perfect matchings
Unions of perfect matchings
how many edges of a bridgeless cubic graph G can be covered
by k perfect matchings?
define mk (G ) as
S
| Mi |
mk (G ) = max
.
M1 ,...,Mk |E (G )|
and set mk to be the infimum of mk (G ) over all bridgeless
cubic G
observe m1 = 1/3
the Berge–Fulkerson conjecture implies that m5 = 1
Tomáš Kaiser
Non-intersecting perfect matchings in cubic graphs
Background
Perfect matchings with sparse intersection
Perfect matchings in cubic graphs
Fano colorings and joins
Unions of perfect matchings
Unions of perfect matchings (cont’d)
Theorem (Kaiser, Král’ and Norine, 2006)
m2 = 3/5 and 27/35 ≤ m3 ≤ 4/5.
proof uses Edmonds’ characterization of the perfect matching
polytope
to prove m3 = 4/5, it would suffice to find two perfect
matchings with sparse intersection — again
Tomáš Kaiser
Non-intersecting perfect matchings in cubic graphs
Background
Perfect matchings with sparse intersection
Perfect matchings in cubic graphs
Fano colorings and joins
Unions of perfect matchings
Unions of perfect matchings (cont’d)
Theorem (Kaiser, Král’ and Norine, 2006)
m2 = 3/5 and 27/35 ≤ m3 ≤ 4/5.
proof uses Edmonds’ characterization of the perfect matching
polytope
to prove m3 = 4/5, it would suffice to find two perfect
matchings with sparse intersection — again
Tomáš Kaiser
Non-intersecting perfect matchings in cubic graphs
Background
Perfect matchings with sparse intersection
A result for graphs of oddness two
Embedded graphs and balanced joins
Proof method
Problems
A result for graphs of oddness two
oddness of a cubic graph G : minimum number of odd cycles
in a 2-factor
Theorem (Kaiser and Raspaud, 2006+)
Every bridgeless cubic graph of oddness two has two perfect
matchings with sparse intersection.
Tomáš Kaiser
Non-intersecting perfect matchings in cubic graphs
Background
Perfect matchings with sparse intersection
A result for graphs of oddness two
Embedded graphs and balanced joins
Proof method
Problems
Contracting the 2-factor
G cubic, M perfect matching in G
orient cycles of G − M
contract each cycle of G − M to a vertex
the resulting graph G̃ (an M-contraction of G ) has a natural
cyclic ordering of edges around each vertex
−→
Tomáš Kaiser
Non-intersecting perfect matchings in cubic graphs
Background
Perfect matchings with sparse intersection
A result for graphs of oddness two
Embedded graphs and balanced joins
Proof method
Problems
Perfect matchings and embedded graphs
the cyclic orderings at vertices form a rotation system
the rotation system determines an embedding of the
M-contraction G̃ in an orientable surface (Heffter; Ringel;
Youngs)
so G̃ can be viewed as an embedded graph
Tomáš Kaiser
Non-intersecting perfect matchings in cubic graphs
Background
Perfect matchings with sparse intersection
A result for graphs of oddness two
Embedded graphs and balanced joins
Proof method
Problems
An example: the Petersen graph
5
4
4
1
3
−→
2
2
1
5
3
5
2
4
1
Embedding in a genus 3 surface:
1
3
2
1 4
Tomáš Kaiser
5
2
Non-intersecting perfect matchings in cubic graphs
Background
Perfect matchings with sparse intersection
A result for graphs of oddness two
Embedded graphs and balanced joins
Proof method
Problems
An example: the Petersen graph
5
4
4
1
3
−→
2
2
1
5
3
5
2
4
1
Embedding in a genus 3 surface:
1
3
2
1 4
Tomáš Kaiser
5
2
Non-intersecting perfect matchings in cubic graphs
Background
Perfect matchings with sparse intersection
A result for graphs of oddness two
Embedded graphs and balanced joins
Proof method
Problems
Balanced joins
Definition
Let H be an embedded graph. A set J ⊂ E (G ) is a balanced join if
at each v ∈ V (G ), each ‘segment’ of edges not in J has even
length.
Observation
A balanced join is a join.
Tomáš Kaiser
Non-intersecting perfect matchings in cubic graphs
Background
Perfect matchings with sparse intersection
A result for graphs of oddness two
Embedded graphs and balanced joins
Proof method
Problems
Balanced joins
Definition
Let H be an embedded graph. A set J ⊂ E (G ) is a balanced join if
at each v ∈ V (G ), each ‘segment’ of edges not in J has even
length.
Observation
A balanced join is a join.
Tomáš Kaiser
Non-intersecting perfect matchings in cubic graphs
Background
Perfect matchings with sparse intersection
A result for graphs of oddness two
Embedded graphs and balanced joins
Proof method
Problems
Joins in the M-contraction
G cubic, M a perfect matching
G̃ is an M-contraction of G
Observation
J̃ is a join in G̃ ⇐⇒ G has a join J with
J ∩ M = J̃.
Observation
J̃ is a balanced join in G̃ ⇐⇒ G has a perfect matching J with
J ∩ M = J̃.
Tomáš Kaiser
Non-intersecting perfect matchings in cubic graphs
Background
Perfect matchings with sparse intersection
A result for graphs of oddness two
Embedded graphs and balanced joins
Proof method
Problems
Joins in the M-contraction
G cubic, M a perfect matching
G̃ is an M-contraction of G
Observation
J̃ is a join in G̃ ⇐⇒ G has a join J with
J ∩ M = J̃.
Observation
J̃ is a balanced join in G̃ ⇐⇒ G has a perfect matching J with
J ∩ M = J̃.
Tomáš Kaiser
Non-intersecting perfect matchings in cubic graphs
Background
Perfect matchings with sparse intersection
A result for graphs of oddness two
Embedded graphs and balanced joins
Proof method
Problems
A reformulation
Recall:
X ⊂ E is sparse if it contains no odd cut
X is sparse ⇐⇒ there is a join J with X ∩ J = ∅
We prove:
Theorem
Every bridgeless embedded graph with only two odd vertices
contains a sparse balanced join.
Tomáš Kaiser
Non-intersecting perfect matchings in cubic graphs
Background
Perfect matchings with sparse intersection
A result for graphs of oddness two
Embedded graphs and balanced joins
Proof method
Problems
A reformulation
Recall:
X ⊂ E is sparse if it contains no odd cut
X is sparse ⇐⇒ there is a join J with X ∩ J = ∅
We prove:
Theorem
Every bridgeless embedded graph with only two odd vertices
contains a sparse balanced join.
Tomáš Kaiser
Non-intersecting perfect matchings in cubic graphs
Background
Perfect matchings with sparse intersection
A result for graphs of oddness two
Embedded graphs and balanced joins
Proof method
Problems
A reformulation (cont’d)
G cubic, M1 a perfect matching, G̃ an M1 -contraction
let J2 be a sparse balanced join in G̃
J2 balanced =⇒ extends to a perfect matching M2 in G with
M1 ∩ M2 = J2
J2 sparse =⇒ G̃ contains a join J 0 disjoint from J2 , this
gives a join J in G with
J ∩ M1 = J 0
we easily get
M1 ∩ M2 ∩ J = ∅
Tomáš Kaiser
Non-intersecting perfect matchings in cubic graphs
Background
Perfect matchings with sparse intersection
A result for graphs of oddness two
Embedded graphs and balanced joins
Proof method
Problems
Splitting
Recall Fleischner’s Splitting Lemma:
Theorem
G 2-edge-connected, v a vertex of degree ≥ 4. Let h0 , h1 , h2 be
edges incident with v . If neither G (v ; h0 , h1 ) nor G (v ; h0 , h2 ) is
2-edge-connected, then {h0 , h1 , h2 } is a 3-cut.
Our main tool is the following variant:
Theorem
Let G have 2 odd degree vertices and let h0 , . . . , h4 be distinct
edges incident with a vertex z of even degree. Assume the other
endvertex of h0 has odd degree. If neither G (z; h1 , h2 ) nor
G (z; h3 , h4 ) is bridgeless, then {h0 , h1 , h2 } or {h0 , h3 , h4 } is a
3-cut.
Tomáš Kaiser
Non-intersecting perfect matchings in cubic graphs
Background
Perfect matchings with sparse intersection
A result for graphs of oddness two
Embedded graphs and balanced joins
Proof method
Problems
Splitting
Recall Fleischner’s Splitting Lemma:
Theorem
G 2-edge-connected, v a vertex of degree ≥ 4. Let h0 , h1 , h2 be
edges incident with v . If neither G (v ; h0 , h1 ) nor G (v ; h0 , h2 ) is
2-edge-connected, then {h0 , h1 , h2 } is a 3-cut.
Our main tool is the following variant:
Theorem
Let G have 2 odd degree vertices and let h0 , . . . , h4 be distinct
edges incident with a vertex z of even degree. Assume the other
endvertex of h0 has odd degree. If neither G (z; h1 , h2 ) nor
G (z; h3 , h4 ) is bridgeless, then {h0 , h1 , h2 } or {h0 , h3 , h4 } is a
3-cut.
Tomáš Kaiser
Non-intersecting perfect matchings in cubic graphs
Background
Perfect matchings with sparse intersection
A result for graphs of oddness two
Embedded graphs and balanced joins
Proof method
Problems
Balanced joins and the dual graph
H embedded graph; a set of faces is independent if it corresponds
to an independent set in H ∗
Proposition
Let F be an independent set of faces in an embedded graph H.
The set of edges of H not incident with any face in F is a balanced
join.
. . . but not all balanced joins arise in this way (consider the
M-contraction of the Petersen graph).
Tomáš Kaiser
Non-intersecting perfect matchings in cubic graphs
Background
Perfect matchings with sparse intersection
A result for graphs of oddness two
Embedded graphs and balanced joins
Proof method
Problems
Balanced joins and the dual graph
H embedded graph; a set of faces is independent if it corresponds
to an independent set in H ∗
Proposition
Let F be an independent set of faces in an embedded graph H.
The set of edges of H not incident with any face in F is a balanced
join.
. . . but not all balanced joins arise in this way (consider the
M-contraction of the Petersen graph).
Tomáš Kaiser
Non-intersecting perfect matchings in cubic graphs
Background
Perfect matchings with sparse intersection
A result for graphs of oddness two
Embedded graphs and balanced joins
Proof method
Problems
Balanced joins and the dual graph
H embedded graph; a set of faces is independent if it corresponds
to an independent set in H ∗
Proposition
Let F be an independent set of faces in an embedded graph H.
The set of edges of H not incident with any face in F is a balanced
join.
. . . but not all balanced joins arise in this way (consider the
M-contraction of the Petersen graph).
Tomáš Kaiser
Non-intersecting perfect matchings in cubic graphs
Background
Perfect matchings with sparse intersection
A result for graphs of oddness two
Embedded graphs and balanced joins
Proof method
Problems
Problems
Definition
The odd edge-connectivity λodd (H) of a graph H = the size of a
smallest odd cut in H.
Conjecture
Let H be an embedded graph with λodd (H) ≥ 5. Then H contains
a sparse balanced join.
Tomáš Kaiser
Non-intersecting perfect matchings in cubic graphs
Background
Perfect matchings with sparse intersection
A result for graphs of oddness two
Embedded graphs and balanced joins
Proof method
Problems
Problems (cont’d)
Problem
Is there an integer k such that every embedded graph H of odd
edge-connectivity λodd (H) ≥ k contains two disjoint balanced
joins?
Problem
Is there a function f such that for every odd `, the edge set of
each embedded graph H with λodd (H) ≥ f (`) can be partitioned
into ` balanced joins?
Tomáš Kaiser
Non-intersecting perfect matchings in cubic graphs
Background
Perfect matchings with sparse intersection
A result for graphs of oddness two
Embedded graphs and balanced joins
Proof method
Problems
Problems (cont’d)
Problem
Is there an integer k such that every embedded graph H of odd
edge-connectivity λodd (H) ≥ k contains two disjoint balanced
joins?
Problem
Is there a function f such that for every odd `, the edge set of
each embedded graph H with λodd (H) ≥ f (`) can be partitioned
into ` balanced joins?
Tomáš Kaiser
Non-intersecting perfect matchings in cubic graphs
Background
Perfect matchings with sparse intersection
A result for graphs of oddness two
Embedded graphs and balanced joins
Proof method
Problems
Thank you for your attention.
Tomáš Kaiser
Non-intersecting perfect matchings in cubic graphs