edge coloring: not so simple on multigraphs? · edge colorings and matchings edge coloring...
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![Page 1: Edge coloring: not so simple on multigraphs? · Edge colorings and matchings Edge coloring ˇDecomposition into matchings. Maximum size of a matching: bjVj 2 c. ˜0(G) max jE(H)j](https://reader036.vdocuments.us/reader036/viewer/2022070922/5fbae3ad07e5ca2b482248a4/html5/thumbnails/1.jpg)
Edge coloring: not so simple on multigraphs?
Marthe Bonamy
May 10, 2012
Marthe Bonamy Edge coloring: not so simple on multigraphs?
![Page 2: Edge coloring: not so simple on multigraphs? · Edge colorings and matchings Edge coloring ˇDecomposition into matchings. Maximum size of a matching: bjVj 2 c. ˜0(G) max jE(H)j](https://reader036.vdocuments.us/reader036/viewer/2022070922/5fbae3ad07e5ca2b482248a4/html5/thumbnails/2.jpg)
Edge coloring
χ: Minimum number of colors to ensure that
a b ⇒ a 6= b.
χ′: Minimum number of colors to ensure that
a b⇒ a 6= b.
∆: Maximum degree of the graph.
∆ ≤ χ′ ≤ 2∆− 1.
Marthe Bonamy Edge coloring: not so simple on multigraphs?
![Page 3: Edge coloring: not so simple on multigraphs? · Edge colorings and matchings Edge coloring ˇDecomposition into matchings. Maximum size of a matching: bjVj 2 c. ˜0(G) max jE(H)j](https://reader036.vdocuments.us/reader036/viewer/2022070922/5fbae3ad07e5ca2b482248a4/html5/thumbnails/3.jpg)
Edge coloring
χ: Minimum number of colors to ensure that
a b ⇒ a 6= b.
χ′: Minimum number of colors to ensure that
a b⇒ a 6= b.
∆: Maximum degree of the graph.
∆ ≤ χ′ ≤ 2∆− 1.
Marthe Bonamy Edge coloring: not so simple on multigraphs?
![Page 4: Edge coloring: not so simple on multigraphs? · Edge colorings and matchings Edge coloring ˇDecomposition into matchings. Maximum size of a matching: bjVj 2 c. ˜0(G) max jE(H)j](https://reader036.vdocuments.us/reader036/viewer/2022070922/5fbae3ad07e5ca2b482248a4/html5/thumbnails/4.jpg)
Edge coloring
χ: Minimum number of colors to ensure that
a b ⇒ a 6= b.
χ′: Minimum number of colors to ensure that
a b⇒ a 6= b.
∆: Maximum degree of the graph.
∆ ≤ χ′ ≤ 2∆− 1.
Marthe Bonamy Edge coloring: not so simple on multigraphs?
![Page 5: Edge coloring: not so simple on multigraphs? · Edge colorings and matchings Edge coloring ˇDecomposition into matchings. Maximum size of a matching: bjVj 2 c. ˜0(G) max jE(H)j](https://reader036.vdocuments.us/reader036/viewer/2022070922/5fbae3ad07e5ca2b482248a4/html5/thumbnails/5.jpg)
Edge coloring
χ: Minimum number of colors to ensure that
a b ⇒ a 6= b.
χ′: Minimum number of colors to ensure that
a b⇒ a 6= b.
∆: Maximum degree of the graph.
∆ ≤ χ′
≤ 2∆− 1.
Marthe Bonamy Edge coloring: not so simple on multigraphs?
![Page 6: Edge coloring: not so simple on multigraphs? · Edge colorings and matchings Edge coloring ˇDecomposition into matchings. Maximum size of a matching: bjVj 2 c. ˜0(G) max jE(H)j](https://reader036.vdocuments.us/reader036/viewer/2022070922/5fbae3ad07e5ca2b482248a4/html5/thumbnails/6.jpg)
Edge coloring
χ: Minimum number of colors to ensure that
a b ⇒ a 6= b.
χ′: Minimum number of colors to ensure that
a b⇒ a 6= b.
∆: Maximum degree of the graph.
∆ ≤ χ′ ≤ 2∆− 1.
Marthe Bonamy Edge coloring: not so simple on multigraphs?
![Page 7: Edge coloring: not so simple on multigraphs? · Edge colorings and matchings Edge coloring ˇDecomposition into matchings. Maximum size of a matching: bjVj 2 c. ˜0(G) max jE(H)j](https://reader036.vdocuments.us/reader036/viewer/2022070922/5fbae3ad07e5ca2b482248a4/html5/thumbnails/7.jpg)
Simple graphs
Theorem (Vizing ’64)
For any simple graph G, ∆(G) ≤ χ′(G ) ≤ ∆(G) +1.
Theorem (Konig ’16, Sanders Zhao ’01)
For any simple graph G, if G is bipartite, or G is planar with∆(G) ≥ 7, then χ′(G ) = ∆(G).
Theorem (Erdos Wilson ’77)
Almost every simple graph G verifies χ′(G ) = ∆(G).
Marthe Bonamy Edge coloring: not so simple on multigraphs?
![Page 8: Edge coloring: not so simple on multigraphs? · Edge colorings and matchings Edge coloring ˇDecomposition into matchings. Maximum size of a matching: bjVj 2 c. ˜0(G) max jE(H)j](https://reader036.vdocuments.us/reader036/viewer/2022070922/5fbae3ad07e5ca2b482248a4/html5/thumbnails/8.jpg)
Simple graphs
Theorem (Vizing ’64)
For any simple graph G, ∆(G) ≤ χ′(G ) ≤ ∆(G) +1.
Theorem (Konig ’16, Sanders Zhao ’01)
For any simple graph G, if G is bipartite, or G is planar with∆(G) ≥ 7, then χ′(G ) = ∆(G).
Theorem (Erdos Wilson ’77)
Almost every simple graph G verifies χ′(G ) = ∆(G).
Marthe Bonamy Edge coloring: not so simple on multigraphs?
![Page 9: Edge coloring: not so simple on multigraphs? · Edge colorings and matchings Edge coloring ˇDecomposition into matchings. Maximum size of a matching: bjVj 2 c. ˜0(G) max jE(H)j](https://reader036.vdocuments.us/reader036/viewer/2022070922/5fbae3ad07e5ca2b482248a4/html5/thumbnails/9.jpg)
Simple graphs
Theorem (Vizing ’64)
For any simple graph G, ∆(G) ≤ χ′(G ) ≤ ∆(G) +1.
Theorem (Holyer ’81)
It is NP-complete to compute χ′ on simple graphs.
Theorem (Misra Gries ’92 (Inspired from the proof of Vizing’stheorem))
For any simple graph G = (V ,E ), a (∆ + 1)-edge-coloring can befound in O(|V | × |E |).
Marthe Bonamy Edge coloring: not so simple on multigraphs?
![Page 10: Edge coloring: not so simple on multigraphs? · Edge colorings and matchings Edge coloring ˇDecomposition into matchings. Maximum size of a matching: bjVj 2 c. ˜0(G) max jE(H)j](https://reader036.vdocuments.us/reader036/viewer/2022070922/5fbae3ad07e5ca2b482248a4/html5/thumbnails/10.jpg)
Simple graphs
Theorem (Vizing ’64)
For any simple graph G, ∆(G) ≤ χ′(G ) ≤ ∆(G) +1.
Theorem (Holyer ’81)
It is NP-complete to compute χ′ on simple graphs.
Theorem (Misra Gries ’92 (Inspired from the proof of Vizing’stheorem))
For any simple graph G = (V ,E ), a (∆ + 1)-edge-coloring can befound in O(|V | × |E |).
Marthe Bonamy Edge coloring: not so simple on multigraphs?
![Page 11: Edge coloring: not so simple on multigraphs? · Edge colorings and matchings Edge coloring ˇDecomposition into matchings. Maximum size of a matching: bjVj 2 c. ˜0(G) max jE(H)j](https://reader036.vdocuments.us/reader036/viewer/2022070922/5fbae3ad07e5ca2b482248a4/html5/thumbnails/11.jpg)
Multigraphs
∆ = 2p.χ′ = 3p.µ = p.
Both theorems areoptimal!
µ: Maximum number of edges sharing the same endpoints.
Theorem (Shannon ’49)
For any multigraph G , χ′(G ) ≤ 3∆(G)2 .
Theorem (Vizing ’64)
For any multigraph G , χ′(G ) ≤ ∆(G ) + µ(G ).
Marthe Bonamy Edge coloring: not so simple on multigraphs?
![Page 12: Edge coloring: not so simple on multigraphs? · Edge colorings and matchings Edge coloring ˇDecomposition into matchings. Maximum size of a matching: bjVj 2 c. ˜0(G) max jE(H)j](https://reader036.vdocuments.us/reader036/viewer/2022070922/5fbae3ad07e5ca2b482248a4/html5/thumbnails/12.jpg)
Multigraphs
∆ = 2p.χ′ = 3p.µ = p.
Both theorems areoptimal!
µ: Maximum number of edges sharing the same endpoints.
Theorem (Shannon ’49)
For any multigraph G , χ′(G ) ≤ 3∆(G)2 .
Theorem (Vizing ’64)
For any multigraph G , χ′(G ) ≤ ∆(G ) + µ(G ).
Marthe Bonamy Edge coloring: not so simple on multigraphs?
![Page 13: Edge coloring: not so simple on multigraphs? · Edge colorings and matchings Edge coloring ˇDecomposition into matchings. Maximum size of a matching: bjVj 2 c. ˜0(G) max jE(H)j](https://reader036.vdocuments.us/reader036/viewer/2022070922/5fbae3ad07e5ca2b482248a4/html5/thumbnails/13.jpg)
Multigraphs
∆ = 2p.χ′ = 3p.µ = p.
Both theorems areoptimal!
µ: Maximum number of edges sharing the same endpoints.
Theorem (Shannon ’49)
For any multigraph G , χ′(G ) ≤ 3∆(G)2 .
Theorem (Vizing ’64)
For any multigraph G , χ′(G ) ≤ ∆(G ) + µ(G ).
Marthe Bonamy Edge coloring: not so simple on multigraphs?
![Page 14: Edge coloring: not so simple on multigraphs? · Edge colorings and matchings Edge coloring ˇDecomposition into matchings. Maximum size of a matching: bjVj 2 c. ˜0(G) max jE(H)j](https://reader036.vdocuments.us/reader036/viewer/2022070922/5fbae3ad07e5ca2b482248a4/html5/thumbnails/14.jpg)
Linear relaxation of edge coloring
M(G ): set of all matchings.w : M(G )→ {0; 1}.
Minimise∑
M w(M) s.t.
∀e ∈ E ,∑
M|e∈M w(M) = 1.
w ′ optimal solution ⇔∑M w ′(M) = χ′(G ).
.
χ′p: Minimum number of colors to ensure that
a1 a2 . . . ap b1 b2 . . . bp
⇒ ∀i , j , ai 6= bj .
Property
χ′f (G ) = infp
χ′p(G)
p .
Marthe Bonamy Edge coloring: not so simple on multigraphs?
![Page 15: Edge coloring: not so simple on multigraphs? · Edge colorings and matchings Edge coloring ˇDecomposition into matchings. Maximum size of a matching: bjVj 2 c. ˜0(G) max jE(H)j](https://reader036.vdocuments.us/reader036/viewer/2022070922/5fbae3ad07e5ca2b482248a4/html5/thumbnails/15.jpg)
Linear relaxation of edge coloring
M(G ): set of all matchings.w : M(G )→ {0; 1}.
Minimise∑
M w(M) s.t.∀e ∈ E ,
∑M|e∈M w(M) = 1.
w ′ optimal solution ⇔∑M w ′(M) = χ′(G ).
.
χ′p: Minimum number of colors to ensure that
a1 a2 . . . ap b1 b2 . . . bp
⇒ ∀i , j , ai 6= bj .
Property
χ′f (G ) = infp
χ′p(G)
p .
Marthe Bonamy Edge coloring: not so simple on multigraphs?
![Page 16: Edge coloring: not so simple on multigraphs? · Edge colorings and matchings Edge coloring ˇDecomposition into matchings. Maximum size of a matching: bjVj 2 c. ˜0(G) max jE(H)j](https://reader036.vdocuments.us/reader036/viewer/2022070922/5fbae3ad07e5ca2b482248a4/html5/thumbnails/16.jpg)
Linear relaxation of edge coloring
M(G ): set of all matchings.w : M(G )→ [0; 1].
Minimise∑
M w(M) s.t.∀e ∈ E ,
∑M|e∈M w(M) = 1.
w ′ optimal solution ⇔∑M w ′(M) = χ′
f (G ).
.
χ′p: Minimum number of colors to ensure that
a1 a2 . . . ap b1 b2 . . . bp
⇒ ∀i , j , ai 6= bj .
Property
χ′f (G ) = infp
χ′p(G)
p .
Marthe Bonamy Edge coloring: not so simple on multigraphs?
![Page 17: Edge coloring: not so simple on multigraphs? · Edge colorings and matchings Edge coloring ˇDecomposition into matchings. Maximum size of a matching: bjVj 2 c. ˜0(G) max jE(H)j](https://reader036.vdocuments.us/reader036/viewer/2022070922/5fbae3ad07e5ca2b482248a4/html5/thumbnails/17.jpg)
Linear relaxation of edge coloring
M(G ): set of all matchings.w : M(G )→ [0; 1].
Minimise∑
M w(M) s.t.∀e ∈ E ,
∑M|e∈M w(M) = 1.
w ′ optimal solution ⇔∑M w ′(M) = χ′
f (G ).
.
χ′p: Minimum number of colors to ensure that
a1 a2 . . . ap b1 b2 . . . bp
⇒ ∀i , j , ai 6= bj .
Property
χ′f (G ) = infp
χ′p(G)
p .
Marthe Bonamy Edge coloring: not so simple on multigraphs?
![Page 18: Edge coloring: not so simple on multigraphs? · Edge colorings and matchings Edge coloring ˇDecomposition into matchings. Maximum size of a matching: bjVj 2 c. ˜0(G) max jE(H)j](https://reader036.vdocuments.us/reader036/viewer/2022070922/5fbae3ad07e5ca2b482248a4/html5/thumbnails/18.jpg)
Linear relaxation of edge coloring
M(G ): set of all matchings.w : M(G )→ [0; 1].
Minimise∑
M w(M) s.t.∀e ∈ E ,
∑M|e∈M w(M) = 1.
w ′ optimal solution ⇔∑M w ′(M) = χ′
f (G ).
.
χ′p: Minimum number of colors to ensure that
a1 a2 . . . ap b1 b2 . . . bp
⇒ ∀i , j , ai 6= bj .
Property
χ′f (G ) = infp
χ′p(G)
p .
Marthe Bonamy Edge coloring: not so simple on multigraphs?
![Page 19: Edge coloring: not so simple on multigraphs? · Edge colorings and matchings Edge coloring ˇDecomposition into matchings. Maximum size of a matching: bjVj 2 c. ˜0(G) max jE(H)j](https://reader036.vdocuments.us/reader036/viewer/2022070922/5fbae3ad07e5ca2b482248a4/html5/thumbnails/19.jpg)
Edge colorings and matchings
Edge coloring ≈ Decomposition into matchings.
Maximum size of a matching: b |V |2 c.
χ′(G ) ≥ max |E(H)|b |V (H)|
2c
= m(G ).
m(G ): density of G
Theorem (Edmonds ’65)
For any multigraph G , χ′f (G ) = max(∆(G),m(G )).
Conjecture (Goldberg ’73)
For any multigraph G , χ′(G ) ≤ max(∆(G) +1, dm(G )e).
Which would be optimal. And is computable in polynomial time.
Marthe Bonamy Edge coloring: not so simple on multigraphs?
![Page 20: Edge coloring: not so simple on multigraphs? · Edge colorings and matchings Edge coloring ˇDecomposition into matchings. Maximum size of a matching: bjVj 2 c. ˜0(G) max jE(H)j](https://reader036.vdocuments.us/reader036/viewer/2022070922/5fbae3ad07e5ca2b482248a4/html5/thumbnails/20.jpg)
Edge colorings and matchings
Edge coloring ≈ Decomposition into matchings.Maximum size of a matching: b |V |
2 c.
χ′(G ) ≥ max |E(H)|b |V (H)|
2c
= m(G ).
m(G ): density of G
Theorem (Edmonds ’65)
For any multigraph G , χ′f (G ) = max(∆(G),m(G )).
Conjecture (Goldberg ’73)
For any multigraph G , χ′(G ) ≤ max(∆(G) +1, dm(G )e).
Which would be optimal. And is computable in polynomial time.
Marthe Bonamy Edge coloring: not so simple on multigraphs?
![Page 21: Edge coloring: not so simple on multigraphs? · Edge colorings and matchings Edge coloring ˇDecomposition into matchings. Maximum size of a matching: bjVj 2 c. ˜0(G) max jE(H)j](https://reader036.vdocuments.us/reader036/viewer/2022070922/5fbae3ad07e5ca2b482248a4/html5/thumbnails/21.jpg)
Edge colorings and matchings
Edge coloring ≈ Decomposition into matchings.Maximum size of a matching: b |V |
2 c.χ′(G ) ≥ max |E(H)|
b |V (H)|2
c= m(G ).
m(G ): density of G
Theorem (Edmonds ’65)
For any multigraph G , χ′f (G ) = max(∆(G),m(G )).
Conjecture (Goldberg ’73)
For any multigraph G , χ′(G ) ≤ max(∆(G) +1, dm(G )e).
Which would be optimal. And is computable in polynomial time.
Marthe Bonamy Edge coloring: not so simple on multigraphs?
![Page 22: Edge coloring: not so simple on multigraphs? · Edge colorings and matchings Edge coloring ˇDecomposition into matchings. Maximum size of a matching: bjVj 2 c. ˜0(G) max jE(H)j](https://reader036.vdocuments.us/reader036/viewer/2022070922/5fbae3ad07e5ca2b482248a4/html5/thumbnails/22.jpg)
Edge colorings and matchings
Edge coloring ≈ Decomposition into matchings.Maximum size of a matching: b |V |
2 c.χ′(G ) ≥ max |E(H)|
b |V (H)|2
c= m(G ).
m(G ): density of G
Theorem (Edmonds ’65)
For any multigraph G , χ′f (G ) = max(∆(G),m(G )).
Conjecture (Goldberg ’73)
For any multigraph G , χ′(G ) ≤ max(∆(G) +1, dm(G )e).
Which would be optimal. And is computable in polynomial time.
Marthe Bonamy Edge coloring: not so simple on multigraphs?
![Page 23: Edge coloring: not so simple on multigraphs? · Edge colorings and matchings Edge coloring ˇDecomposition into matchings. Maximum size of a matching: bjVj 2 c. ˜0(G) max jE(H)j](https://reader036.vdocuments.us/reader036/viewer/2022070922/5fbae3ad07e5ca2b482248a4/html5/thumbnails/23.jpg)
Edge colorings and matchings
Edge coloring ≈ Decomposition into matchings.Maximum size of a matching: b |V |
2 c.χ′(G ) ≥ max |E(H)|
b |V (H)|2
c= m(G ).
m(G ): density of G
Theorem (Edmonds ’65)
For any multigraph G , χ′f (G ) = max(∆(G),m(G )).
Conjecture (Goldberg ’73)
For any multigraph G , χ′(G ) ≤ max(∆(G) +1, dm(G )e).
Which would be optimal.
And is computable in polynomial time.
Marthe Bonamy Edge coloring: not so simple on multigraphs?
![Page 24: Edge coloring: not so simple on multigraphs? · Edge colorings and matchings Edge coloring ˇDecomposition into matchings. Maximum size of a matching: bjVj 2 c. ˜0(G) max jE(H)j](https://reader036.vdocuments.us/reader036/viewer/2022070922/5fbae3ad07e5ca2b482248a4/html5/thumbnails/24.jpg)
Edge colorings and matchings
Edge coloring ≈ Decomposition into matchings.Maximum size of a matching: b |V |
2 c.χ′(G ) ≥ max |E(H)|
b |V (H)|2
c= m(G ).
m(G ): density of G
Theorem (Edmonds ’65)
For any multigraph G , χ′f (G ) = max(∆(G),m(G )).
Conjecture (Goldberg ’73)
For any multigraph G , χ′(G ) ≤ max(∆(G) +1, dm(G )e).
Which would be optimal. And is computable in polynomial time.
Marthe Bonamy Edge coloring: not so simple on multigraphs?
![Page 25: Edge coloring: not so simple on multigraphs? · Edge colorings and matchings Edge coloring ˇDecomposition into matchings. Maximum size of a matching: bjVj 2 c. ˜0(G) max jE(H)j](https://reader036.vdocuments.us/reader036/viewer/2022070922/5fbae3ad07e5ca2b482248a4/html5/thumbnails/25.jpg)
Goldberg’s conjecture
Theorem (Yu ’08)
For any multigraph G , χ′(G ) ≤ max(∆(G) +√
∆(G)2 ,w(G )).
Theorem (Kurt ’09)
For any multigraph G , χ′(G ) ≤ max(∆(G) + ∆(G) +2224 ,w(G )).
Marthe Bonamy Edge coloring: not so simple on multigraphs?
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Seymour’s conjecture
Conjecture (Seymour)
For any k-regular planar multigraph G s.t. every odd subset H hasd(H) ≥ k verifies χ′(G ) = k.
k = 3⇔ 4CT. (Tait)
k = 4, 5 (Guenin)
k = 6 (Dvorak, Kawarabayashi, Kral)
k = 7 (Edwards, Kawarabayashi)
k = 8 (Chudnovsky, Edwards, Seymour)
Sketch of the proof.
Marthe Bonamy Edge coloring: not so simple on multigraphs?
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Seymour’s conjecture
Conjecture (Seymour)
For any k-regular planar multigraph G s.t. every odd subset H hasd(H) ≥ k verifies χ′(G ) = k.
k = 3⇔ 4CT. (Tait)
k = 4, 5 (Guenin)
k = 6 (Dvorak, Kawarabayashi, Kral)
k = 7 (Edwards, Kawarabayashi)
k = 8 (Chudnovsky, Edwards, Seymour)
Sketch of the proof.
Marthe Bonamy Edge coloring: not so simple on multigraphs?
![Page 28: Edge coloring: not so simple on multigraphs? · Edge colorings and matchings Edge coloring ˇDecomposition into matchings. Maximum size of a matching: bjVj 2 c. ˜0(G) max jE(H)j](https://reader036.vdocuments.us/reader036/viewer/2022070922/5fbae3ad07e5ca2b482248a4/html5/thumbnails/28.jpg)
Conclusion
Conjecture (Jensen Toft ’95)
For any simple graph G on an even number of vertices,χ′(G ) = ∆(G) or χ′(G ) = ∆(G ).
Thanks for your attention.Any questions?
Marthe Bonamy Edge coloring: not so simple on multigraphs?
![Page 29: Edge coloring: not so simple on multigraphs? · Edge colorings and matchings Edge coloring ˇDecomposition into matchings. Maximum size of a matching: bjVj 2 c. ˜0(G) max jE(H)j](https://reader036.vdocuments.us/reader036/viewer/2022070922/5fbae3ad07e5ca2b482248a4/html5/thumbnails/29.jpg)
Conclusion
Conjecture (Jensen Toft ’95)
For any simple graph G on an even number of vertices,χ′(G ) = ∆(G) or χ′(G ) = ∆(G ).
Thanks for your attention.Any questions?
Marthe Bonamy Edge coloring: not so simple on multigraphs?