Edge chromatic number–Upper bounds χ e ( G ) ≤ 3∆ (Shannon, 1949). 2 χ e ( G ) ≤ ∆ + µ where µ is the multiplicity of G (Vizing, 1964). χ e ( G ) ≤ ∆ + ⌈ ∆ − 2 g o − 1 ⌉ (Goldberg, 1970s). χ e ( G ) ≤ ∆ + ⌈ µ 2 ⌋ ⌉ (Stephen, 2001). ⌊ g In general χ e ( G ) ∈ { ∆ , ∆ + 1 , · · · , ∆ + µ } . Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 8 / 1
Edge chromatic number–Upper bounds χ e ( G ) ≤ 3∆ (Shannon, 1949). 2 χ e ( G ) ≤ ∆ + µ where µ is the multiplicity of G (Vizing, 1964). χ e ( G ) ≤ ∆ + ⌈ ∆ − 2 g o − 1 ⌉ (Goldberg, 1970s). χ e ( G ) ≤ ∆ + ⌈ µ 2 ⌋ ⌉ (Stephen, 2001). ⌊ g In general χ e ( G ) ∈ { ∆ , ∆ + 1 , · · · , ∆ + µ } . For a simple graph G , χ e ( G ) ∈ { ∆ , ∆ + 1 } . Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 8 / 1
Edge chromatic number–Another nontrivial lower bound Suppose G has an edge k -coloring with the color classes E 1 , E 2 , · · · E k where k = χ e ( G ) Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 9 / 1
Edge chromatic number–Another nontrivial lower bound Suppose G has an edge k -coloring with the color classes E 1 , E 2 , · · · E k where k = χ e ( G ) Each color class is a matching, so | E i | ≤ ⌊ | V ( G ) | ⌋ . 2 Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 9 / 1
Edge chromatic number–Another nontrivial lower bound Suppose G has an edge k -coloring with the color classes E 1 , E 2 , · · · E k where k = χ e ( G ) Each color class is a matching, so | E i | ≤ ⌊ | V ( G ) | ⌋ . 2 The total number of edges in G , | E ( G ) | ≤ k ⌊ | V ( G ) | ⌋ . 2 Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 9 / 1
Edge chromatic number–Another nontrivial lower bound Suppose G has an edge k -coloring with the color classes E 1 , E 2 , · · · E k where k = χ e ( G ) Each color class is a matching, so | E i | ≤ ⌊ | V ( G ) | ⌋ . 2 The total number of edges in G , | E ( G ) | ≤ k ⌊ | V ( G ) | ⌋ . 2 χ e ( G ) = k ≥ ⌈ | E ( G ) | ⌋ ⌉ . ⌊ | V ( G ) | 2 Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 9 / 1
Edge chromatic number–Another nontrivial lower bound Suppose G has an edge k -coloring with the color classes E 1 , E 2 , · · · E k where k = χ e ( G ) Each color class is a matching, so | E i | ≤ ⌊ | V ( G ) | ⌋ . 2 The total number of edges in G , | E ( G ) | ≤ k ⌊ | V ( G ) | ⌋ . 2 χ e ( G ) = k ≥ ⌈ | E ( G ) | ⌋ ⌉ . ⌊ | V ( G ) | 2 For any subgraph H of G , we have χ e ( G ) ≥ χ e ( H ) ≥ ⌈ | E ( H ) | ⌋ ⌉ . ⌊ | V ( H ) | 2 Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 9 / 1
Edge chromatic number–Another nontrivial lower bound Suppose G has an edge k -coloring with the color classes E 1 , E 2 , · · · E k where k = χ e ( G ) Each color class is a matching, so | E i | ≤ ⌊ | V ( G ) | ⌋ . 2 The total number of edges in G , | E ( G ) | ≤ k ⌊ | V ( G ) | ⌋ . 2 χ e ( G ) = k ≥ ⌈ | E ( G ) | ⌋ ⌉ . ⌊ | V ( G ) | 2 For any subgraph H of G , we have χ e ( G ) ≥ χ e ( H ) ≥ ⌈ | E ( H ) | ⌋ ⌉ . ⌊ | V ( H ) | 2 H ⊆ G , | V ( H ) |≥ 2 ⌈ | E ( H ) | χ e ( G ) ≥ max ⌉ ⌊ | V ( H ) | ⌋ 2 Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 9 / 1
Edge chromatic number–Another nontrivial lower bound H ⊆ G , | V ( H ) |≥ 2 ⌈ | E ( H ) | χ e ( G ) ≥ max ⌉ ⌊ | V ( H ) |⌋ 2 Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 10 / 1
Edge chromatic number–Another nontrivial lower bound H ⊆ G , | V ( H ) |≥ 2 ⌈ | E ( H ) | χ e ( G ) ≥ max ⌉ ⌊ | V ( H ) |⌋ 2 H ⊆ G , | V ( H ) |≥ 2 ⌈ | E ( H ) | w ( G ) = max ⌉ ⌊ | V ( H ) | ⌋ 2 Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 10 / 1
Edge chromatic number–Another nontrivial lower bound H ⊆ G , | V ( H ) |≥ 2 ⌈ | E ( H ) | χ e ( G ) ≥ max ⌉ ⌊ | V ( H ) |⌋ 2 H ⊆ G , | V ( H ) |≥ 2 ⌈ | E ( H ) | w ( G ) = max ⌉ ⌊ | V ( H ) | ⌋ 2 w ( G ) is called the density of G . Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 10 / 1
Edge chromatic number–Another nontrivial lower bound H ⊆ G , | V ( H ) |≥ 2 ⌈ | E ( H ) | χ e ( G ) ≥ max ⌉ ⌊ | V ( H ) |⌋ 2 H ⊆ G , | V ( H ) |≥ 2 ⌈ | E ( H ) | w ( G ) = max ⌉ ⌊ | V ( H ) | ⌋ 2 w ( G ) is called the density of G . χ e ( G ) ≥ w ( G ) Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 10 / 1
Edge chromatic number–Another nontrivial lower bound H ⊆ G , | V ( H ) |≥ 2 ⌈ | E ( H ) | χ e ( G ) ≥ max ⌉ ⌊ | V ( H ) |⌋ 2 H ⊆ G , | V ( H ) |≥ 2 ⌈ | E ( H ) | w ( G ) = max ⌉ ⌊ | V ( H ) | ⌋ 2 w ( G ) is called the density of G . χ e ( G ) ≥ w ( G ) χ e ( G ) ≥ max { ∆ , w ( G ) } . Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 10 / 1
Edge chromatic number–Another nontrivial lower bound H ⊆ G , | V ( H ) |≥ 2 ⌈ | E ( H ) | χ e ( G ) ≥ max ⌉ ⌊ | V ( H ) |⌋ 2 H ⊆ G , | V ( H ) |≥ 2 ⌈ | E ( H ) | w ( G ) = max ⌉ ⌊ | V ( H ) | ⌋ 2 w ( G ) is called the density of G . χ e ( G ) ≥ w ( G ) χ e ( G ) ≥ max { ∆ , w ( G ) } . The maximum value can be achieved when | V ( H ) | is odd. Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 10 / 1
Edge chromatic number–Another nontrivial lower bound H ⊆ G , | V ( H ) |≥ 2 ⌈ | E ( H ) | χ e ( G ) ≥ max ⌉ ⌊ | V ( H ) |⌋ 2 H ⊆ G , | V ( H ) |≥ 2 ⌈ | E ( H ) | w ( G ) = max ⌉ ⌊ | V ( H ) | ⌋ 2 w ( G ) is called the density of G . χ e ( G ) ≥ w ( G ) χ e ( G ) ≥ max { ∆ , w ( G ) } . The maximum value can be achieved when | V ( H ) | is odd. H ⊆ G , | V ( H ) |≥ 2 ⌈ | E ( H ) | H ⊆ G , | V ( H ) |≥ 3 , odd ⌈ 2 | E ( H ) | w ( G ) = max ⌉ = max | V ( H ) | − 1 ⌉ ⌊ | V ( H ) | ⌋ 2 Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 10 / 1
Seymour’s r -graph Conjecture For simple graphs, ∆ ≤ χ e ( G ) ≤ ∆ + 1 . Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 11 / 1
Seymour’s r -graph Conjecture For simple graphs, ∆ ≤ χ e ( G ) ≤ ∆ + 1 . In general, χ e ( G ) ≥ max { ∆ , w ( G ) } . Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 11 / 1
Seymour’s r -graph Conjecture For simple graphs, ∆ ≤ χ e ( G ) ≤ ∆ + 1 . In general, χ e ( G ) ≥ max { ∆ , w ( G ) } . Is it true χ e ( G ) ≤ max { ∆ , w ( G ) } + 1 . ? Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 11 / 1
Seymour’s r -graph Conjecture For simple graphs, ∆ ≤ χ e ( G ) ≤ ∆ + 1 . In general, χ e ( G ) ≥ max { ∆ , w ( G ) } . Is it true χ e ( G ) ≤ max { ∆ , w ( G ) } + 1 . ? Conjecture (Seymour’s r-graph Conjecture, 1979) Let G be a graph. Then χ e ( G ) ≤ max { ∆ , w ( G ) } + 1 . Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 11 / 1
Goldberg Conjecture Conjecture (Goldberg Conjecture) Let G be a graph. Then χ e ( G ) ≤ max { ∆ + 1 , w ( G ) } . Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 12 / 1
Goldberg Conjecture Conjecture (Goldberg Conjecture) Let G be a graph. Then χ e ( G ) ≤ max { ∆ + 1 , w ( G ) } . Goldberg conjecture was proposed by Goldberg in 1970 and independently by Seymour in 1979. Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 12 / 1
Goldberg Conjecture Conjecture (Goldberg Conjecture) Let G be a graph. Then χ e ( G ) ≤ max { ∆ + 1 , w ( G ) } . Goldberg conjecture was proposed by Goldberg in 1970 and independently by Seymour in 1979. Goldberg Conjecture is equivalent to the following statements. Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 12 / 1
Goldberg Conjecture Conjecture (Goldberg Conjecture) Let G be a graph. Then χ e ( G ) ≤ max { ∆ + 1 , w ( G ) } . Goldberg conjecture was proposed by Goldberg in 1970 and independently by Seymour in 1979. Goldberg Conjecture is equivalent to the following statements. Let G be a graph. If χ e ( G ) ≥ ∆ + 2, then χ e ( G ) = w ( G ) . Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 12 / 1
Goldberg Conjecture Conjecture (Goldberg Conjecture) Let G be a graph. Then χ e ( G ) ≤ max { ∆ + 1 , w ( G ) } . Goldberg conjecture was proposed by Goldberg in 1970 and independently by Seymour in 1979. Goldberg Conjecture is equivalent to the following statements. Let G be a graph. If χ e ( G ) ≥ ∆ + 2, then χ e ( G ) = w ( G ) . χ e ( G ) ∈ { ∆ , ∆ + 1 , w ( G ) } . Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 12 / 1
Goldberg Conjecture–Importance Goldberg Conjecture implies that if χ e ( G ) ≥ ∆ + 2, then there is a polynomial algorithm to compute χ e ( G ). Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 13 / 1
Goldberg Conjecture–Importance Goldberg Conjecture implies that if χ e ( G ) ≥ ∆ + 2, then there is a polynomial algorithm to compute χ e ( G ). So it implies that the difficulty in determining χ e ( G ) is only to distinguish between two cases χ e ( G ) = ∆ and χ e ( G ) = ∆ + 1, which is NP-hard proved by Holyer in 1980. Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 13 / 1
Goldberg Conjecture–Importance Goldberg Conjecture implies that if χ e ( G ) ≥ ∆ + 2, then there is a polynomial algorithm to compute χ e ( G ). So it implies that the difficulty in determining χ e ( G ) is only to distinguish between two cases χ e ( G ) = ∆ and χ e ( G ) = ∆ + 1, which is NP-hard proved by Holyer in 1980. Goldberg Conjecture also implies Seymour’s r -graph conjecture and Jakobsen’s critical graph conjecture. Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 13 / 1
Goldberg Conjecture–Importance Goldberg Conjecture implies that if χ e ( G ) ≥ ∆ + 2, then there is a polynomial algorithm to compute χ e ( G ). So it implies that the difficulty in determining χ e ( G ) is only to distinguish between two cases χ e ( G ) = ∆ and χ e ( G ) = ∆ + 1, which is NP-hard proved by Holyer in 1980. Goldberg Conjecture also implies Seymour’s r -graph conjecture and Jakobsen’s critical graph conjecture. G is critical if χ e ( G − e ) < χ e ( G ) for any edge e . Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 13 / 1
Seymour’s r -graph Conjecture–Original Version A r -regular graph is an r -graph if for every set X ⊆ V ( G ) with | X | odd, | ∂ G ( X ) | ≥ r . Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 14 / 1
Seymour’s r -graph Conjecture–Original Version A r -regular graph is an r -graph if for every set X ⊆ V ( G ) with | X | odd, | ∂ G ( X ) | ≥ r . Conjecture (Seymour’s r-graph Conjecture, 1979) Every r-graph satisfies χ e ( G ) ≤ r + 1 . Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 14 / 1
Seymour’s r -graph Conjecture–Original Version A r -regular graph is an r -graph if for every set X ⊆ V ( G ) with | X | odd, | ∂ G ( X ) | ≥ r . Conjecture (Seymour’s r-graph Conjecture, 1979) Every r-graph satisfies χ e ( G ) ≤ r + 1 . w ( G ) ≤ r for each r -graph. Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 14 / 1
Seymour’s r -graph Conjecture–Original Version A r -regular graph is an r -graph if for every set X ⊆ V ( G ) with | X | odd, | ∂ G ( X ) | ≥ r . Conjecture (Seymour’s r-graph Conjecture, 1979) Every r-graph satisfies χ e ( G ) ≤ r + 1 . w ( G ) ≤ r for each r -graph. Let H ⊆ G with | V ( H ) | odd. Let X = V ( H ). Then 2 | E ( H ) | ≤ r | X | − r = r ( | X | − 1) Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 14 / 1
Seymour’s r -graph Conjecture–Original Version A r -regular graph is an r -graph if for every set X ⊆ V ( G ) with | X | odd, | ∂ G ( X ) | ≥ r . Conjecture (Seymour’s r-graph Conjecture, 1979) Every r-graph satisfies χ e ( G ) ≤ r + 1 . w ( G ) ≤ r for each r -graph. Let H ⊆ G with | V ( H ) | odd. Let X = V ( H ). Then 2 | E ( H ) | ≤ r | X | − r = r ( | X | − 1) | V ( H ) |− 1 ≤ r ( | X |− 1) 2 | E ( H ) | = r . | X |− 1 Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 14 / 1
Seymour’s r -graph Conjecture–Original Version A r -regular graph is an r -graph if for every set X ⊆ V ( G ) with | X | odd, | ∂ G ( X ) | ≥ r . Conjecture (Seymour’s r-graph Conjecture, 1979) Every r-graph satisfies χ e ( G ) ≤ r + 1 . w ( G ) ≤ r for each r -graph. Let H ⊆ G with | V ( H ) | odd. Let X = V ( H ). Then 2 | E ( H ) | ≤ r | X | − r = r ( | X | − 1) | V ( H ) |− 1 ≤ r ( | X |− 1) 2 | E ( H ) | = r . | X |− 1 If an r -regular graph has an edge r -coloring, then it must be an r -graph. (Why?) Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 14 / 1
Seymour’s r -graph Conjecture–Original Version A r -regular graph is an r -graph if for every set X ⊆ V ( G ) with | X | odd, | ∂ G ( X ) | ≥ r . Conjecture (Seymour’s r-graph Conjecture, 1979) Every r-graph satisfies χ e ( G ) ≤ r + 1 . w ( G ) ≤ r for each r -graph. Let H ⊆ G with | V ( H ) | odd. Let X = V ( H ). Then 2 | E ( H ) | ≤ r | X | − r = r ( | X | − 1) | V ( H ) |− 1 ≤ r ( | X |− 1) 2 | E ( H ) | = r . | X |− 1 If an r -regular graph has an edge r -coloring, then it must be an r -graph. (Why?) Seymour proved that every graph with max { ∆ , w ( G ) } ≤ r is contained in an r -graph as a subgraph. Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 14 / 1
Seymour’s r -graph Conjecture–Original Version A r -regular graph is an r -graph if for every set X ⊆ V ( G ) with | X | odd, | ∂ G ( X ) | ≥ r . Conjecture (Seymour’s r-graph Conjecture, 1979) Every r-graph satisfies χ e ( G ) ≤ r + 1 . w ( G ) ≤ r for each r -graph. Let H ⊆ G with | V ( H ) | odd. Let X = V ( H ). Then 2 | E ( H ) | ≤ r | X | − r = r ( | X | − 1) | V ( H ) |− 1 ≤ r ( | X |− 1) 2 | E ( H ) | = r . | X |− 1 If an r -regular graph has an edge r -coloring, then it must be an r -graph. (Why?) Seymour proved that every graph with max { ∆ , w ( G ) } ≤ r is contained in an r -graph as a subgraph. Since w ( G ) ≤ r for each r -graph, Goldberg Conjecture implies χ e ( G ) ≤ max { ∆ + 1 , w ( G ) } = r + 1. Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 14 / 1
Seymour’s r -graph Conjecture–Equivalent Version Seymour’s r -graph conjecture is equivalent to Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 15 / 1
Seymour’s r -graph Conjecture–Equivalent Version Seymour’s r -graph conjecture is equivalent to Every graph G satisfies χ e ( G ) ≤ max { ∆ , w ( G ) } + 1. Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 15 / 1
Seymour’s r -graph Conjecture–Equivalent Version Seymour’s r -graph conjecture is equivalent to Every graph G satisfies χ e ( G ) ≤ max { ∆ , w ( G ) } + 1. Seymour’s r -graph conjecture is true for r ≤ 15. Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 15 / 1
Seymour’s r -graph Conjecture–Equivalent Version Seymour’s r -graph conjecture is equivalent to Every graph G satisfies χ e ( G ) ≤ max { ∆ , w ( G ) } + 1. Seymour’s r -graph conjecture is true for r ≤ 15. Seymour’s r -graph conjecture suggests that if G is an r -graph then, for all t ≥ 1, either χ e ( tG ) = max { ∆( tG ) , w ( tG ) } or χ e ( tG ) = max { ∆( tG ) , w ( tG ) } + 1. Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 15 / 1
Seymour’s Exact Conjecture For planar graph, Seymour proposed a stronger conjecture. Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 16 / 1
Seymour’s Exact Conjecture For planar graph, Seymour proposed a stronger conjecture. Conjecture (Seymour’s Exact Conjecture, 1979) Every planar graph G satisfies χ e ( G ) = max { ∆ , w ( G ) } . Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 16 / 1
Seymour’s Exact Conjecture For planar graph, Seymour proposed a stronger conjecture. Conjecture (Seymour’s Exact Conjecture, 1979) Every planar graph G satisfies χ e ( G ) = max { ∆ , w ( G ) } . The cases r = 0 , 1 , 2 are trivial. Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 16 / 1
Seymour’s Exact Conjecture For planar graph, Seymour proposed a stronger conjecture. Conjecture (Seymour’s Exact Conjecture, 1979) Every planar graph G satisfies χ e ( G ) = max { ∆ , w ( G ) } . The cases r = 0 , 1 , 2 are trivial. Seymour proved the exact conjecture for series-parrallel graphs in 1990. Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 16 / 1
Seymour’s Exact Conjecture For planar graph, Seymour proposed a stronger conjecture. Conjecture (Seymour’s Exact Conjecture, 1979) Every planar graph G satisfies χ e ( G ) = max { ∆ , w ( G ) } . The cases r = 0 , 1 , 2 are trivial. Seymour proved the exact conjecture for series-parrallel graphs in 1990. Marcotte verified the conjecture for the class of graphs not containing K 3 , 3 or K − 5 as a minor, 2001. Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 16 / 1
Seymour’s Exact Conjecture For planar graph, Seymour proposed a stronger conjecture. Conjecture (Seymour’s Exact Conjecture, 1979) Every planar graph G satisfies χ e ( G ) = max { ∆ , w ( G ) } . The cases r = 0 , 1 , 2 are trivial. Seymour proved the exact conjecture for series-parrallel graphs in 1990. Marcotte verified the conjecture for the class of graphs not containing K 3 , 3 or K − 5 as a minor, 2001. The case r = 3 is equivalent to the Four Color Theorem. Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 16 / 1
Seymour’s Exact Conjecture For planar graph, Seymour proposed a stronger conjecture. Conjecture (Seymour’s Exact Conjecture, 1979) Every planar graph G satisfies χ e ( G ) = max { ∆ , w ( G ) } . The cases r = 0 , 1 , 2 are trivial. Seymour proved the exact conjecture for series-parrallel graphs in 1990. Marcotte verified the conjecture for the class of graphs not containing K 3 , 3 or K − 5 as a minor, 2001. The case r = 3 is equivalent to the Four Color Theorem. The cases r = 4 , 5 was proved by Guenin (2011). Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 16 / 1
Seymour’s Exact Conjecture For planar graph, Seymour proposed a stronger conjecture. Conjecture (Seymour’s Exact Conjecture, 1979) Every planar graph G satisfies χ e ( G ) = max { ∆ , w ( G ) } . The cases r = 0 , 1 , 2 are trivial. Seymour proved the exact conjecture for series-parrallel graphs in 1990. Marcotte verified the conjecture for the class of graphs not containing K 3 , 3 or K − 5 as a minor, 2001. The case r = 3 is equivalent to the Four Color Theorem. The cases r = 4 , 5 was proved by Guenin (2011). Dvorak, Kawarabayashi, and Kral solved the case r = 6 in 2011 Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 16 / 1
Seymour’s Exact Conjecture For planar graph, Seymour proposed a stronger conjecture. Conjecture (Seymour’s Exact Conjecture, 1979) Every planar graph G satisfies χ e ( G ) = max { ∆ , w ( G ) } . The cases r = 0 , 1 , 2 are trivial. Seymour proved the exact conjecture for series-parrallel graphs in 1990. Marcotte verified the conjecture for the class of graphs not containing K 3 , 3 or K − 5 as a minor, 2001. The case r = 3 is equivalent to the Four Color Theorem. The cases r = 4 , 5 was proved by Guenin (2011). Dvorak, Kawarabayashi, and Kral solved the case r = 6 in 2011 The case of k = 7 was proved by Edwards and Kawarabayashi in 2012. Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 16 / 1
Seymour’s Exact Conjecture For planar graph, Seymour proposed a stronger conjecture. Conjecture (Seymour’s Exact Conjecture, 1979) Every planar graph G satisfies χ e ( G ) = max { ∆ , w ( G ) } . The cases r = 0 , 1 , 2 are trivial. Seymour proved the exact conjecture for series-parrallel graphs in 1990. Marcotte verified the conjecture for the class of graphs not containing K 3 , 3 or K − 5 as a minor, 2001. The case r = 3 is equivalent to the Four Color Theorem. The cases r = 4 , 5 was proved by Guenin (2011). Dvorak, Kawarabayashi, and Kral solved the case r = 6 in 2011 The case of k = 7 was proved by Edwards and Kawarabayashi in 2012. Chudnovsky, Edwards and Seymour solved the case k = 8 in 2012. Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 16 / 1
Jakobsen’s critical graph conjecture Conjecture (Jakobosen’s Critical Graph Conjecture, 1973) Let G be a critical graph, and let m − 1∆( G ) + m − 3 m χ e ( G ) > m − 1 . for an odd integer m ≥ 3 . Then | V ( G ) | ≤ m − 2 . Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 17 / 1
Jakobsen’s critical graph conjecture Conjecture (Jakobosen’s Critical Graph Conjecture, 1973) Let G be a critical graph, and let m − 1∆( G ) + m − 3 m χ e ( G ) > m − 1 . for an odd integer m ≥ 3 . Then | V ( G ) | ≤ m − 2 . Jakobsen’s conjecture is trivial for m = 3 by Shannons upper bound. Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 17 / 1
Jakobsen’s critical graph conjecture Conjecture (Jakobosen’s Critical Graph Conjecture, 1973) Let G be a critical graph, and let m − 1∆( G ) + m − 3 m χ e ( G ) > m − 1 . for an odd integer m ≥ 3 . Then | V ( G ) | ≤ m − 2 . Jakobsen’s conjecture is trivial for m = 3 by Shannons upper bound. Anderson proved that Goldberg’s Conjecture implies Jakobsen’s conjecture. Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 17 / 1
Jakobsen’s critical graph conjecture Conjecture (Jakobosen’s Critical Graph Conjecture, 1973) Let G be a critical graph, and let m − 1∆( G ) + m − 3 m χ e ( G ) > m − 1 . for an odd integer m ≥ 3 . Then | V ( G ) | ≤ m − 2 . Jakobsen’s conjecture is trivial for m = 3 by Shannons upper bound. Anderson proved that Goldberg’s Conjecture implies Jakobsen’s conjecture. Jakobsen’s conjecture was verified for 5 ≤ m ≤ 15. Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 17 / 1
Fractional edge chromatic number An edge coloring can be considered as an Integer Programming Problem. Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 18 / 1
Fractional edge chromatic number An edge coloring can be considered as an Integer Programming Problem. Let M denote the set of all matchings of a graph G . Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 18 / 1
Fractional edge chromatic number An edge coloring can be considered as an Integer Programming Problem. Let M denote the set of all matchings of a graph G . For each edge e , let M e denote the set of all matchings containing e . Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 18 / 1
Fractional edge chromatic number An edge coloring can be considered as an Integer Programming Problem. Let M denote the set of all matchings of a graph G . For each edge e , let M e denote the set of all matchings containing e . � χ e ( G ) = min y M , M ∈M subject to: (1) � M ∈ M e y M = 1 for each edge e ∈ E ( G ). (2) y M ∈ { 0 , 1 } Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 18 / 1
Fractional chromatic index The fractional edge chromatic number χ ∗ e ( G ) is defined as: Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 19 / 1
Fractional chromatic index The fractional edge chromatic number χ ∗ e ( G ) is defined as: χ ∗ � e ( G ) = min y M , M ∈M subject to: (1) � M ∈ M e y M = 1 for each edge e ∈ E ( G ). (2) 0 ≤ y M ≤ 1 Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 19 / 1
Fractional chromatic index The fractional edge chromatic number χ ∗ e ( G ) is defined as: χ ∗ � e ( G ) = min y M , M ∈M subject to: (1) � M ∈ M e y M = 1 for each edge e ∈ E ( G ). (2) 0 ≤ y M ≤ 1 With matching techniques one can compute the fractional edge chromatic number in polynomial time. Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 19 / 1
Fractional chromatic index The fractional edge chromatic number χ ∗ e ( G ) is defined as: χ ∗ � e ( G ) = min y M , M ∈M subject to: (1) � M ∈ M e y M = 1 for each edge e ∈ E ( G ). (2) 0 ≤ y M ≤ 1 With matching techniques one can compute the fractional edge chromatic number in polynomial time. From Edmond’s matching polytope theorem, | E ( H ) | χ ∗ e ( G ) = max { ∆ , max 2 | V ( H ) |⌋} . ⌊ 1 H ⊆ G , | V ( H ) |≥ 2 Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 19 / 1
Fractional chromatic index The fractional edge chromatic number χ ∗ e ( G ) is defined as: χ ∗ � e ( G ) = min y M , M ∈M subject to: (1) � M ∈ M e y M = 1 for each edge e ∈ E ( G ). (2) 0 ≤ y M ≤ 1 With matching techniques one can compute the fractional edge chromatic number in polynomial time. From Edmond’s matching polytope theorem, | E ( H ) | χ ∗ e ( G ) = max { ∆ , max 2 | V ( H ) |⌋} . ⌊ 1 H ⊆ G , | V ( H ) |≥ 2 If χ ∗ e ( G ) = ∆, then w ( G ) ≤ ∆ Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 19 / 1
Fractional chromatic index The fractional edge chromatic number χ ∗ e ( G ) is defined as: χ ∗ � e ( G ) = min y M , M ∈M subject to: (1) � M ∈ M e y M = 1 for each edge e ∈ E ( G ). (2) 0 ≤ y M ≤ 1 With matching techniques one can compute the fractional edge chromatic number in polynomial time. From Edmond’s matching polytope theorem, | E ( H ) | χ ∗ e ( G ) = max { ∆ , max 2 | V ( H ) |⌋} . ⌊ 1 H ⊆ G , | V ( H ) |≥ 2 If χ ∗ e ( G ) = ∆, then w ( G ) ≤ ∆ If χ ∗ e ( G ) > ∆, then w ( G ) = ⌈ χ ∗ e ( G ) ⌉ and w ( G ) can be computed in polynomial time. Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 19 / 1
Fractional chromatic index The computation of the edge chromatic number χ e ( G ) is NP-hard Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 20 / 1
Fractional chromatic index The computation of the edge chromatic number χ e ( G ) is NP-hard The fractional edge chromatic number can be computed in polynomial time. Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 20 / 1
Fractional chromatic index The computation of the edge chromatic number χ e ( G ) is NP-hard The fractional edge chromatic number can be computed in polynomial time. It is not clear whether the density w ( G ) can be computed in polynomial time. Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 20 / 1
Fractional chromatic index The computation of the edge chromatic number χ e ( G ) is NP-hard The fractional edge chromatic number can be computed in polynomial time. It is not clear whether the density w ( G ) can be computed in polynomial time. max { ∆( G ) , w ( G ) } can be computed in polynomial time Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 20 / 1
Fractional chromatic index The computation of the edge chromatic number χ e ( G ) is NP-hard The fractional edge chromatic number can be computed in polynomial time. It is not clear whether the density w ( G ) can be computed in polynomial time. max { ∆( G ) , w ( G ) } can be computed in polynomial time Goldberg Conjecture is equivalent to the claim that χ e ( G ) = ⌈ χ ∗ e ( G ) ⌉ for every graph G with χ e ( G ) ≥ ∆ + 2. Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 20 / 1
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