Edge coloring: not so simple on multigraphs?

Marthe Bonamy

May 10, 2012

Marthe Bonamy Edge coloring: not so simple on multigraphs? χ: Minimum number of colors to ensure that

a b ⇒ a 6= b.

χ0: Minimum number of colors to ensure that

a b ⇒ a 6= b.

∆: Maximum degree of the graph.

∆ ≤ χ0 ≤ 2∆ − 1.

Edge coloring

Marthe Bonamy Edge coloring: not so simple on multigraphs? χ0: Minimum number of colors to ensure that

a b ⇒ a 6= b.

∆: Maximum degree of the graph.

∆ ≤ χ0 ≤ 2∆ − 1.

Edge coloring

χ: Minimum number of colors to ensure that

a b ⇒ a 6= b.

Marthe Bonamy Edge coloring: not so simple on multigraphs? ∆: Maximum degree of the graph.

∆ ≤ χ0 ≤ 2∆ − 1.

Edge coloring

χ: Minimum number of colors to ensure that

a b ⇒ a 6= b.

χ0: Minimum number of colors to ensure that

a b ⇒ a 6= b.

Marthe Bonamy Edge coloring: not so simple on multigraphs? ≤ 2∆ − 1.

Edge coloring

χ: Minimum number of colors to ensure that

a b ⇒ a 6= b.

χ0: Minimum number of colors to ensure that

a b ⇒ a 6= b.

∆: Maximum degree of the graph.

∆ ≤ χ0

Marthe Bonamy Edge coloring: not so simple on multigraphs? Edge coloring

χ: Minimum number of colors to ensure that

a b ⇒ a 6= b.

χ0: Minimum number of colors to ensure that

a b ⇒ a 6= b.

∆: Maximum degree of the graph.

∆ ≤ χ0 ≤ 2∆ − 1.

Marthe Bonamy Edge coloring: not so simple on multigraphs? Theorem (K¨onig’16, Sanders Zhao ’01) For any simple graph G, if G is bipartite, or G is planar with ∆(G) ≥ 7, then χ0(G) = ∆(G).

Theorem (Erd¨osWilson ’77) Almost every simple graph G verifies χ0(G) = ∆(G).

Simple graphs

Theorem (Vizing ’64) For any simple graph G, ∆(G) ≤ χ0(G) ≤ ∆(G) +1.

Marthe Bonamy Edge coloring: not so simple on multigraphs? Simple graphs

Theorem (Vizing ’64) For any simple graph G, ∆(G) ≤ χ0(G) ≤ ∆(G) +1.

Theorem (K¨onig’16, Sanders Zhao ’01) For any simple graph G, if G is bipartite, or G is planar with ∆(G) ≥ 7, then χ0(G) = ∆(G).

Theorem (Erd¨osWilson ’77) Almost every simple graph G verifies χ0(G) = ∆(G).

Marthe Bonamy Edge coloring: not so simple on multigraphs? Theorem (Misra Gries ’92 (Inspired from the proof of Vizing’s theorem)) For any simple graph G = (V , E), a (∆ + 1)-edge-coloring can be found in O(|V | × |E|).

Simple graphs

Theorem (Vizing ’64) For any simple graph G, ∆(G) ≤ χ0(G) ≤ ∆(G) +1.

Theorem (Holyer ’81) It is NP-complete to compute χ0 on simple graphs.

Marthe Bonamy Edge coloring: not so simple on multigraphs? Simple graphs

Theorem (Vizing ’64) For any simple graph G, ∆(G) ≤ χ0(G) ≤ ∆(G) +1.

Theorem (Holyer ’81) It is NP-complete to compute χ0 on simple graphs.

Theorem (Misra Gries ’92 (Inspired from the proof of Vizing’s theorem)) For any simple graph G = (V , E), a (∆ + 1)-edge-coloring can be found in O(|V | × |E|).

Marthe Bonamy Edge coloring: not so simple on multigraphs? Both theorems are optimal!

Theorem (Shannon ’49)

0 3∆(G) For any multigraph G, χ (G) ≤ 2 .

Theorem (Vizing ’64) For any multigraph G, χ0(G) ≤ ∆(G) + µ(G).

Multigraphs

∆ = 2p. χ0 = 3p. µ = p.

µ: Maximum number of edges sharing the same endpoints.

Marthe Bonamy Edge coloring: not so simple on multigraphs? Both theorems are optimal!

Multigraphs

∆ = 2p. χ0 = 3p. µ = p.

µ: Maximum number of edges sharing the same endpoints. Theorem (Shannon ’49)

0 3∆(G) For any multigraph G, χ (G) ≤ 2 .

Theorem (Vizing ’64) For any multigraph G, χ0(G) ≤ ∆(G) + µ(G).

Marthe Bonamy Edge coloring: not so simple on multigraphs? Multigraphs

∆ = 2p. Both theorems are 0 χ = 3p. optimal! µ = p.

µ: Maximum number of edges sharing the same endpoints. Theorem (Shannon ’49)

0 3∆(G) For any multigraph G, χ (G) ≤ 2 .

Theorem (Vizing ’64) For any multigraph G, χ0(G) ≤ ∆(G) + µ(G).

Marthe Bonamy Edge coloring: not so simple on multigraphs? w 0 optimal solution ⇔ P 0 0 P M w (M) = χ (G). Minimise M w(M) s.t. .

0 χp: Minimum number of colors to ensure that

a1 a2 ... ap b1 b2 ... bp ⇒ ∀i, j, ai 6= bj .

Property 0 0 χp(G) χf (G) = infp p .

Linear relaxation of edge coloring

M(G): set of all matchings. w : M(G) → {0; 1}.

P ∀e ∈ E, M|e∈M w(M) = 1.

Marthe Bonamy Edge coloring: not so simple on multigraphs? .

0 χp: Minimum number of colors to ensure that

a1 a2 ... ap b1 b2 ... bp ⇒ ∀i, j, ai 6= bj .

Property 0 0 χp(G) χf (G) = infp p .

Linear relaxation of edge coloring

M(G): set of all matchings. w 0 optimal solution ⇔ w : M(G) → {0; 1}. P 0 0 P M w (M) = χ (G). Minimise M w(M) s.t. P ∀e ∈ E, M|e∈M w(M) = 1.

Marthe Bonamy Edge coloring: not so simple on multigraphs? .

0 χp: Minimum number of colors to ensure that

a1 a2 ... ap b1 b2 ... bp ⇒ ∀i, j, ai 6= bj .

Property 0 0 χp(G) χf (G) = infp p .

Linear relaxation of edge coloring

M(G): set of all matchings. w 0 optimal solution ⇔ w : M(G) → [0; 1]. P 0 0 P M w (M) = χf (G). Minimise M w(M) s.t. P ∀e ∈ E, M|e∈M w(M) = 1.

Marthe Bonamy Edge coloring: not so simple on multigraphs? .

Property 0 0 χp(G) χf (G) = infp p .

Linear relaxation of edge coloring

M(G): set of all matchings. w 0 optimal solution ⇔ w : M(G) → [0; 1]. P 0 0 P M w (M) = χf (G). Minimise M w(M) s.t. P ∀e ∈ E, M|e∈M w(M) = 1.

0 χp: Minimum number of colors to ensure that

a1 a2 ... ap b1 b2 ... bp ⇒ ∀i, j, ai 6= bj .

Marthe Bonamy Edge coloring: not so simple on multigraphs? Linear relaxation of edge coloring

M(G): set of all matchings. w 0 optimal solution ⇔ w : M(G) → [0; 1]. P 0 0 P M w (M) = χf (G). Minimise M w(M) s.t. ∀e ∈ E, P w(M) = 1. . M|e∈M

0 χp: Minimum number of colors to ensure that

a1 a2 ... ap b1 b2 ... bp ⇒ ∀i, j, ai 6= bj .

Property 0 0 χp(G) χf (G) = infp p .

Marthe Bonamy Edge coloring: not so simple on multigraphs? |V | Maximum size of a matching: b 2 c. 0 |E(H)| χ (G) ≥ max |V (H)| = m(G). b 2 c m(G): density of G

Theorem (Edmonds ’65) 0 For any multigraph G, χf (G) = max(∆(G), m(G)).

Conjecture (Goldberg ’73) For any multigraph G, χ0(G) ≤ max(∆(G) +1, dm(G)e). Which would be optimal. And is computable in polynomial time.

Edge colorings and matchings

Edge coloring ≈ Decomposition into matchings.

Marthe Bonamy Edge coloring: not so simple on multigraphs? 0 |E(H)| χ (G) ≥ max |V (H)| = m(G). b 2 c m(G): density of G

Theorem (Edmonds ’65) 0 For any multigraph G, χf (G) = max(∆(G), m(G)).

Conjecture (Goldberg ’73) For any multigraph G, χ0(G) ≤ max(∆(G) +1, dm(G)e). Which would be optimal. And is computable in polynomial time.

Edge colorings and matchings

Edge coloring ≈ Decomposition into matchings. |V | Maximum size of a matching: b 2 c.

Marthe Bonamy Edge coloring: not so simple on multigraphs? Theorem (Edmonds ’65) 0 For any multigraph G, χf (G) = max(∆(G), m(G)).

Conjecture (Goldberg ’73) For any multigraph G, χ0(G) ≤ max(∆(G) +1, dm(G)e). Which would be optimal. And is computable in polynomial time.

Edge colorings and matchings

Edge coloring ≈ Decomposition into matchings. |V | Maximum size of a matching: b 2 c. 0 |E(H)| χ (G) ≥ max |V (H)| = m(G). b 2 c m(G): density of G

Marthe Bonamy Edge coloring: not so simple on multigraphs? Conjecture (Goldberg ’73) For any multigraph G, χ0(G) ≤ max(∆(G) +1, dm(G)e). Which would be optimal. And is computable in polynomial time.

Edge colorings and matchings

Edge coloring ≈ Decomposition into matchings. |V | Maximum size of a matching: b 2 c. 0 |E(H)| χ (G) ≥ max |V (H)| = m(G). b 2 c m(G): density of G

Theorem (Edmonds ’65) 0 For any multigraph G, χf (G) = max(∆(G), m(G)).

Marthe Bonamy Edge coloring: not so simple on multigraphs? And is computable in polynomial time.

Edge colorings and matchings

Edge coloring ≈ Decomposition into matchings. |V | Maximum size of a matching: b 2 c. 0 |E(H)| χ (G) ≥ max |V (H)| = m(G). b 2 c m(G): density of G

Theorem (Edmonds ’65) 0 For any multigraph G, χf (G) = max(∆(G), m(G)).

Conjecture (Goldberg ’73) For any multigraph G, χ0(G) ≤ max(∆(G) +1, dm(G)e). Which would be optimal.

Marthe Bonamy Edge coloring: not so simple on multigraphs? Edge colorings and matchings

Edge coloring ≈ Decomposition into matchings. |V | Maximum size of a matching: b 2 c. 0 |E(H)| χ (G) ≥ max |V (H)| = m(G). b 2 c m(G): density of G

Theorem (Edmonds ’65) 0 For any multigraph G, χf (G) = max(∆(G), m(G)).

Conjecture (Goldberg ’73) For any multigraph G, χ0(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? Goldberg’s conjecture

Theorem (Yu ’08) q 0 ∆(G) For any multigraph G, χ (G) ≤ max(∆(G) + 2 , w(G)).

Theorem (Kurt ’09)

0 ∆(G) +22 For any multigraph G, χ (G) ≤ max(∆(G) + 24 , w(G)).

Marthe Bonamy Edge coloring: not so simple on multigraphs? Sketch of the proof.

Seymour’s conjecture

Conjecture (Seymour) For any k-regular planar multigraph G s.t. every odd subset H has d(H) ≥ k verifies χ0(G) = k.

k = 3 ⇔ 4CT. (Tait) k = 4, 5 (Guenin) k = 6 (Dvorak, Kawarabayashi, Kral) k = 7 (Edwards, Kawarabayashi) k = 8 (Chudnovsky, Edwards, Seymour)

Marthe Bonamy Edge coloring: not so simple on multigraphs? Seymour’s conjecture

Conjecture (Seymour) For any k-regular planar multigraph G s.t. every odd subset H has d(H) ≥ k verifies χ0(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? Thanks for your attention. Any questions?

Conclusion

Conjecture (Jensen Toft ’95) For any simple graph G on an even number of vertices, χ0(G) = ∆(G) or χ0(G) = ∆(G).

Marthe Bonamy Edge coloring: not so simple on multigraphs? Conclusion

Conjecture (Jensen Toft ’95) For any simple graph G on an even number of vertices, χ0(G) = ∆(G) or χ0(G) = ∆(G).

Thanks for your attention. Any questions?

Marthe Bonamy Edge coloring: not so simple on multigraphs?