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(jV j × jEj). 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(jV j × jEj). 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 ) 8i; 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) ! f0; 1g. P 8e 2 E; Mje2M 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 ) 8i; 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) ! f0; 1g. P 0 0 P M w (M) = χ (G). Minimise M w(M) s.t. P 8e 2 E; Mje2M 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 ) 8i; 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 8e 2 E; Mje2M 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 8e 2 E; Mje2M w(M) = 1. 0 χp: Minimum number of colors to ensure that a1 a2 ::: ap b1 b2 ::: bp ) 8i; 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. 8e 2 E; P w(M) = 1. Mje2M 0 χp: Minimum number of colors to ensure that a1 a2 ::: ap b1 b2 ::: bp ) 8i; j; ai 6= bj . Property 0 0 χp(G) χf (G) = infp p . Marthe Bonamy Edge coloring: not so simple on multigraphs? jV j Maximum size of a matching: b 2 c. 0 jE(H)j χ (G) ≥ max jV (H)j = 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 jE(H)j χ (G) ≥ max jV (H)j = 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. jV j 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. jV j Maximum size of a matching: b 2 c. 0 jE(H)j χ (G) ≥ max jV (H)j = 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. jV j Maximum size of a matching: b 2 c. 0 jE(H)j χ (G) ≥ max jV (H)j = 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. jV j Maximum size of a matching: b 2 c. 0 jE(H)j χ (G) ≥ max jV (H)j = 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. jV j Maximum size of a matching: b 2 c. 0 jE(H)j χ (G) ≥ max jV (H)j = 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)).