Some Pretty Edge Coloring Conjectures Rong Luo Department of - - PowerPoint PPT Presentation
Some Pretty Edge Coloring Conjectures Rong Luo Department of Mathematics West Virginia University Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 1 / 1 Introduction A simple graph is a graph without multiple edges while a graph means
Some Pretty Edge Coloring Conjectures
Rong Luo
Department of Mathematics West Virginia University
Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 1 / 1
Introduction
A simple graph is a graph without multiple edges while a graph means parallel edges allowed.
Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 2 / 1
Introduction
A simple graph is a graph without multiple edges while a graph means parallel edges allowed. An edge coloring of a graph is a function assigning values (colors) to the edges of the graph in such a way that any two adjacent edges receive different colors.
Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 2 / 1
Introduction
A simple graph is a graph without multiple edges while a graph means parallel edges allowed. An edge coloring of a graph is a function assigning values (colors) to the edges of the graph in such a way that any two adjacent edges receive different colors. The central problem in this area is to determine the minimum number
Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 2 / 1
Introduction
A simple graph is a graph without multiple edges while a graph means parallel edges allowed. An edge coloring of a graph is a function assigning values (colors) to the edges of the graph in such a way that any two adjacent edges receive different colors. The central problem in this area is to determine the minimum number
This is called the edge chromatic number, denoted χe = χe(G).
Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 2 / 1
Introduction
A simple graph is a graph without multiple edges while a graph means parallel edges allowed. An edge coloring of a graph is a function assigning values (colors) to the edges of the graph in such a way that any two adjacent edges receive different colors. The central problem in this area is to determine the minimum number
This is called the edge chromatic number, denoted χe = χe(G). χe ≥ ∆.
Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 2 / 1
Introduction
A simple graph is a graph without multiple edges while a graph means parallel edges allowed. An edge coloring of a graph is a function assigning values (colors) to the edges of the graph in such a way that any two adjacent edges receive different colors. The central problem in this area is to determine the minimum number
This is called the edge chromatic number, denoted χe = χe(G). χe ≥ ∆. How high can edge chromatic number be?
Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 2 / 1
History
Edge coloring first appeared in graph theory literature in the 1880’s.
Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 3 / 1
History
Edge coloring first appeared in graph theory literature in the 1880’s. In 1880, Tait was attempting to prove the Four Color Theorem, and had shown
Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 3 / 1
History
Edge coloring first appeared in graph theory literature in the 1880’s. In 1880, Tait was attempting to prove the Four Color Theorem, and had shown
Theorem
(Tait) The Four Color Theorem is equivalent to that every 2-edge connected cubic planar graph is edge-3-colorable.
Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 3 / 1
History
Edge coloring first appeared in graph theory literature in the 1880’s. In 1880, Tait was attempting to prove the Four Color Theorem, and had shown
Theorem
(Tait) The Four Color Theorem is equivalent to that every 2-edge connected cubic planar graph is edge-3-colorable.
Theorem
(Tait) Every 2-edge connected cubic planar graph is edge 3-colorable.
Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 3 / 1
History
Edge coloring first appeared in graph theory literature in the 1880’s. In 1880, Tait was attempting to prove the Four Color Theorem, and had shown
Theorem
(Tait) The Four Color Theorem is equivalent to that every 2-edge connected cubic planar graph is edge-3-colorable.
Theorem
(Tait) Every 2-edge connected cubic planar graph is edge 3-colorable. Found to be flawed by Petersen in 1891.
Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 3 / 1
History
Edge coloring first appeared in graph theory literature in the 1880’s. In 1880, Tait was attempting to prove the Four Color Theorem, and had shown
Theorem
(Tait) The Four Color Theorem is equivalent to that every 2-edge connected cubic planar graph is edge-3-colorable.
Theorem
(Tait) Every 2-edge connected cubic planar graph is edge 3-colorable. Found to be flawed by Petersen in 1891. His results stimulated interest in edge-coloring.
Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 3 / 1
History
Chapter 9, Twenty Pretty Edge Coloring Conjectures
Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 4 / 1
History
Chapter 9, Twenty Pretty Edge Coloring Conjectures
9 (late 1960s), 6 (1970s), 4 (early 1980s), 1 (1990)
Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 4 / 1
History
Tashkinov, 2000, introduced a method, called Tashkinov tree, which is a generalization of Vizing Fan and Kiearstead path.
Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 5 / 1
History
Tashkinov, 2000, introduced a method, called Tashkinov tree, which is a generalization of Vizing Fan and Kiearstead path. Three Ph.D dissertations on edge coloring (Goldberg Conjecture)
Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 5 / 1
History
Tashkinov, 2000, introduced a method, called Tashkinov tree, which is a generalization of Vizing Fan and Kiearstead path. Three Ph.D dissertations on edge coloring (Goldberg Conjecture) Kurt O. (2009) The Ohio State University, (Neil Robertson)
Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 5 / 1
History
Tashkinov, 2000, introduced a method, called Tashkinov tree, which is a generalization of Vizing Fan and Kiearstead path. Three Ph.D dissertations on edge coloring (Goldberg Conjecture) Kurt O. (2009) The Ohio State University, (Neil Robertson) MacDonald, J. (2009), University of Waterloo (Penny Haxell)
Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 5 / 1
History
Tashkinov, 2000, introduced a method, called Tashkinov tree, which is a generalization of Vizing Fan and Kiearstead path. Three Ph.D dissertations on edge coloring (Goldberg Conjecture) Kurt O. (2009) The Ohio State University, (Neil Robertson) MacDonald, J. (2009), University of Waterloo (Penny Haxell) Scheide (2007), ( Michael Stiebitz)
Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 5 / 1
Edge chromatic number
χe(G) ≥ ∆.
Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 6 / 1
Edge chromatic number
χe(G) ≥ ∆. How high can edge chromatic number be?
Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 6 / 1
Edge chromatic number
χe(G) ≥ ∆. How high can edge chromatic number be?
Theorem
(Konig, 1916) If G is a bipartite graph, then χe(G) = ∆.
Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 6 / 1
Edge chromatic number
χe(G) ≥ ∆. How high can edge chromatic number be?
Theorem
(Konig, 1916) If G is a bipartite graph, then χe(G) = ∆. Konig’s result is the first result on edge coloring with a correct proof.
Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 6 / 1
Edge chromatic number
χe(G) ≥ ∆. How high can edge chromatic number be?
Theorem
(Konig, 1916) If G is a bipartite graph, then χe(G) = ∆. Konig’s result is the first result on edge coloring with a correct proof.
Theorem
(Vizing Theorem) For each simple graph G, χe(G) = ∆ or ∆ + 1.
Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 6 / 1
Edge chromatic number
χe(G) ≥ ∆. How high can edge chromatic number be?
Theorem
(Konig, 1916) If G is a bipartite graph, then χe(G) = ∆. Konig’s result is the first result on edge coloring with a correct proof.
Theorem
(Vizing Theorem) For each simple graph G, χe(G) = ∆ or ∆ + 1. For a simple graph G, χe(G) ∈ {∆, ∆ + 1}.
Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 6 / 1
Edge chromatic number
Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 7 / 1
Edge chromatic number–Upper bounds
χe(G) ≤ 3∆
2
(Shannon, 1949).
Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 8 / 1
Edge chromatic number–Upper bounds
χe(G) ≤ 3∆
2
(Shannon, 1949). χe(G) ≤ ∆ + µ where µ is the multiplicity of G (Vizing, 1964).
Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 8 / 1
Edge chromatic number–Upper bounds
χe(G) ≤ 3∆
2
(Shannon, 1949). χe(G) ≤ ∆ + µ where µ is the multiplicity of G (Vizing, 1964). χe(G) ≤ ∆ + ⌈ ∆−2
go−1⌉
(Goldberg, 1970s).
Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 8 / 1
Edge chromatic number–Upper bounds
χe(G) ≤ 3∆
2
(Shannon, 1949). χe(G) ≤ ∆ + µ where µ is the multiplicity of G (Vizing, 1964). χe(G) ≤ ∆ + ⌈ ∆−2
go−1⌉
(Goldberg, 1970s). χe(G) ≤ ∆ + ⌈ µ
⌊ g
2 ⌋⌉
(Stephen, 2001).
Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 8 / 1
Edge chromatic number–Upper bounds
χe(G) ≤ 3∆
2
(Shannon, 1949). χe(G) ≤ ∆ + µ where µ is the multiplicity of G (Vizing, 1964). χe(G) ≤ ∆ + ⌈ ∆−2
go−1⌉
(Goldberg, 1970s). χe(G) ≤ ∆ + ⌈ µ
⌊ g
2 ⌋⌉
(Stephen, 2001). In general χe(G) ∈ {∆, ∆ + 1, · · · , ∆ + µ}.
Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 8 / 1
Edge chromatic number–Upper bounds
χe(G) ≤ 3∆
2
(Shannon, 1949). χe(G) ≤ ∆ + µ where µ is the multiplicity of G (Vizing, 1964). χe(G) ≤ ∆ + ⌈ ∆−2
go−1⌉
(Goldberg, 1970s). χe(G) ≤ ∆ + ⌈ µ
⌊ g
2 ⌋⌉
(Stephen, 2001). In general χe(G) ∈ {∆, ∆ + 1, · · · , ∆ + µ}. For a simple graph G, χe(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 E1, E2, · · · Ek where k = χe(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 E1, E2, · · · Ek where k = χe(G) Each color class is a matching, so |Ei| ≤ ⌊|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 E1, E2, · · · Ek where k = χe(G) Each color class is a matching, so |Ei| ≤ ⌊|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 E1, E2, · · · Ek where k = χe(G) Each color class is a matching, so |Ei| ≤ ⌊|V (G)|
2
⌋. The total number of edges in G , |E(G)| ≤ k⌊ |V (G)|
2
⌋. χe(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 E1, E2, · · · Ek where k = χe(G) Each color class is a matching, so |Ei| ≤ ⌊|V (G)|
2
⌋. The total number of edges in G , |E(G)| ≤ k⌊ |V (G)|
2
⌋. χe(G) = k ≥ ⌈ |E(G)|
⌊ |V (G)|
2
⌋⌉.
For any subgraph H of G, we have χe(G) ≥ χe(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 E1, E2, · · · Ek where k = χe(G) Each color class is a matching, so |Ei| ≤ ⌊|V (G)|
2
⌋. The total number of edges in G , |E(G)| ≤ k⌊ |V (G)|
2
⌋. χe(G) = k ≥ ⌈ |E(G)|
⌊ |V (G)|
2
⌋⌉.
For any subgraph H of G, we have χe(G) ≥ χe(H) ≥ ⌈ |E(H)|
⌊ |V (H)|
2
⌋⌉.
χe(G) ≥ max
H⊆G,|V (H)|≥2⌈ |E(H)|
⌊ |V (H)|
2
⌋ ⌉
Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 9 / 1
Edge chromatic number–Another nontrivial lower bound
χe(G) ≥ max
H⊆G,|V (H)|≥2⌈ |E(H)|
⌊ |V (H)
2
|⌋ ⌉
Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 10 / 1
Edge chromatic number–Another nontrivial lower bound
χe(G) ≥ max
H⊆G,|V (H)|≥2⌈ |E(H)|
⌊ |V (H)
2
|⌋ ⌉ w(G) = max
H⊆G,|V (H)|≥2⌈ |E(H)|
⌊ |V (H)|
2
⌋ ⌉
Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 10 / 1
Edge chromatic number–Another nontrivial lower bound
χe(G) ≥ max
H⊆G,|V (H)|≥2⌈ |E(H)|
⌊ |V (H)
2
|⌋ ⌉ w(G) = max
H⊆G,|V (H)|≥2⌈ |E(H)|
⌊ |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
χe(G) ≥ max
H⊆G,|V (H)|≥2⌈ |E(H)|
⌊ |V (H)
2
|⌋ ⌉ w(G) = max
H⊆G,|V (H)|≥2⌈ |E(H)|
⌊ |V (H)|
2
⌋ ⌉ w(G) is called the density of G. χe(G) ≥ w(G)
Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 10 / 1
Edge chromatic number–Another nontrivial lower bound
χe(G) ≥ max
H⊆G,|V (H)|≥2⌈ |E(H)|
⌊ |V (H)
2
|⌋ ⌉ w(G) = max
H⊆G,|V (H)|≥2⌈ |E(H)|
⌊ |V (H)|
2
⌋ ⌉ w(G) is called the density of G. χe(G) ≥ w(G) χe(G) ≥ max{∆, w(G)}.
Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 10 / 1
Edge chromatic number–Another nontrivial lower bound
χe(G) ≥ max
H⊆G,|V (H)|≥2⌈ |E(H)|
⌊ |V (H)
2
|⌋ ⌉ w(G) = max
H⊆G,|V (H)|≥2⌈ |E(H)|
⌊ |V (H)|
2
⌋ ⌉ w(G) is called the density of G. χe(G) ≥ w(G) χe(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
χe(G) ≥ max
H⊆G,|V (H)|≥2⌈ |E(H)|
⌊ |V (H)
2
|⌋ ⌉ w(G) = max
H⊆G,|V (H)|≥2⌈ |E(H)|
⌊ |V (H)|
2
⌋ ⌉ w(G) is called the density of G. χe(G) ≥ w(G) χe(G) ≥ max{∆, w(G)}. The maximum value can be achieved when |V (H)| is odd. w(G) = max
H⊆G,|V (H)|≥2⌈ |E(H)|
⌊ |V (H)|
2
⌋ ⌉ = max
H⊆G,|V (H)|≥3,odd⌈ 2|E(H)|
|V (H)| − 1⌉
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Seymour’s r-graph Conjecture
For simple graphs, ∆ ≤ χe(G) ≤ ∆ + 1.
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Seymour’s r-graph Conjecture
For simple graphs, ∆ ≤ χe(G) ≤ ∆ + 1. In general, χe(G) ≥ max{∆, w(G)}.
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Seymour’s r-graph Conjecture
For simple graphs, ∆ ≤ χe(G) ≤ ∆ + 1. In general, χe(G) ≥ max{∆, w(G)}. Is it true χe(G) ≤ max{∆, w(G)} + 1.?
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Seymour’s r-graph Conjecture
For simple graphs, ∆ ≤ χe(G) ≤ ∆ + 1. In general, χe(G) ≥ max{∆, w(G)}. Is it true χe(G) ≤ max{∆, w(G)} + 1.?
Conjecture
(Seymour’s r-graph Conjecture, 1979) Let G be a graph. Then χe(G) ≤ max{∆, w(G)} + 1.
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Goldberg Conjecture
Conjecture
(Goldberg Conjecture) Let G be a graph. Then χe(G) ≤ max{∆ + 1, w(G)}.
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Goldberg Conjecture
Conjecture
(Goldberg Conjecture) Let G be a graph. Then χe(G) ≤ max{∆ + 1, w(G)}. Goldberg conjecture was proposed by Goldberg in 1970 and independently by Seymour in 1979.
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Goldberg Conjecture
Conjecture
(Goldberg Conjecture) Let G be a graph. Then χe(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.
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Goldberg Conjecture
Conjecture
(Goldberg Conjecture) Let G be a graph. Then χe(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 χe(G) ≥ ∆ + 2, then χe(G) = w(G).
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Goldberg Conjecture
Conjecture
(Goldberg Conjecture) Let G be a graph. Then χe(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 χe(G) ≥ ∆ + 2, then χe(G) = w(G). χe(G) ∈ {∆, ∆ + 1, w(G)}.
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Goldberg Conjecture–Importance
Goldberg Conjecture implies that if χe(G) ≥ ∆ + 2, then there is a polynomial algorithm to compute χe(G).
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Goldberg Conjecture–Importance
Goldberg Conjecture implies that if χe(G) ≥ ∆ + 2, then there is a polynomial algorithm to compute χe(G). So it implies that the difficulty in determining χe(G) is only to distinguish between two cases χe(G) = ∆ and χe(G) = ∆ + 1, which is NP-hard proved by Holyer in 1980.
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Goldberg Conjecture–Importance
Goldberg Conjecture implies that if χe(G) ≥ ∆ + 2, then there is a polynomial algorithm to compute χe(G). So it implies that the difficulty in determining χe(G) is only to distinguish between two cases χe(G) = ∆ and χe(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.
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Goldberg Conjecture–Importance
Goldberg Conjecture implies that if χe(G) ≥ ∆ + 2, then there is a polynomial algorithm to compute χe(G). So it implies that the difficulty in determining χe(G) is only to distinguish between two cases χe(G) = ∆ and χe(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 χe(G − e) < χe(G) for any edge e.
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Seymour’s r-graph Conjecture–Original Version
A r-regular graph is an r-graph if for every set X ⊆ V (G) with |X|
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Seymour’s r-graph Conjecture–Original Version
A r-regular graph is an r-graph if for every set X ⊆ V (G) with |X|
Conjecture
(Seymour’s r-graph Conjecture, 1979) Every r-graph satisfies χe(G) ≤ r + 1.
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Seymour’s r-graph Conjecture–Original Version
A r-regular graph is an r-graph if for every set X ⊆ V (G) with |X|
Conjecture
(Seymour’s r-graph Conjecture, 1979) Every r-graph satisfies χe(G) ≤ r + 1. w(G) ≤ r for each r-graph.
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Seymour’s r-graph Conjecture–Original Version
A r-regular graph is an r-graph if for every set X ⊆ V (G) with |X|
Conjecture
(Seymour’s r-graph Conjecture, 1979) Every r-graph satisfies χe(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)
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Seymour’s r-graph Conjecture–Original Version
A r-regular graph is an r-graph if for every set X ⊆ V (G) with |X|
Conjecture
(Seymour’s r-graph Conjecture, 1979) Every r-graph satisfies χe(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)
2|E(H)| |V (H)|−1 ≤ r(|X|−1) |X|−1
= r.
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Seymour’s r-graph Conjecture–Original Version
A r-regular graph is an r-graph if for every set X ⊆ V (G) with |X|
Conjecture
(Seymour’s r-graph Conjecture, 1979) Every r-graph satisfies χe(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)
2|E(H)| |V (H)|−1 ≤ r(|X|−1) |X|−1
= r. If an r-regular graph has an edge r-coloring, then it must be an r-graph. (Why?)
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Seymour’s r-graph Conjecture–Original Version
A r-regular graph is an r-graph if for every set X ⊆ V (G) with |X|
Conjecture
(Seymour’s r-graph Conjecture, 1979) Every r-graph satisfies χe(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)
2|E(H)| |V (H)|−1 ≤ r(|X|−1) |X|−1
= r. 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.
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Seymour’s r-graph Conjecture–Original Version
A r-regular graph is an r-graph if for every set X ⊆ V (G) with |X|
Conjecture
(Seymour’s r-graph Conjecture, 1979) Every r-graph satisfies χe(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)
2|E(H)| |V (H)|−1 ≤ r(|X|−1) |X|−1
= r. 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 χe(G) ≤ max{∆ + 1, w(G)} = r + 1.
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Seymour’s r-graph Conjecture–Equivalent Version
Seymour’s r-graph conjecture is equivalent to
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Seymour’s r-graph Conjecture–Equivalent Version
Seymour’s r-graph conjecture is equivalent to Every graph G satisfies χe(G) ≤ max{∆, w(G)} + 1.
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Seymour’s r-graph Conjecture–Equivalent Version
Seymour’s r-graph conjecture is equivalent to Every graph G satisfies χe(G) ≤ max{∆, w(G)} + 1. Seymour’s r-graph conjecture is true for r ≤ 15.
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Seymour’s r-graph Conjecture–Equivalent Version
Seymour’s r-graph conjecture is equivalent to Every graph G satisfies χe(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 χe(tG) = max{∆(tG), w(tG)} or χe(tG) = max{∆(tG), w(tG)} + 1.
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Seymour’s Exact Conjecture
For planar graph, Seymour proposed a stronger conjecture.
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Seymour’s Exact Conjecture
For planar graph, Seymour proposed a stronger conjecture.
Conjecture
(Seymour’s Exact Conjecture, 1979) Every planar graph G satisfies χe(G) = max{∆, w(G)}.
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Seymour’s Exact Conjecture
For planar graph, Seymour proposed a stronger conjecture.
Conjecture
(Seymour’s Exact Conjecture, 1979) Every planar graph G satisfies χe(G) = max{∆, w(G)}. The cases r = 0, 1, 2 are trivial.
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Seymour’s Exact Conjecture
For planar graph, Seymour proposed a stronger conjecture.
Conjecture
(Seymour’s Exact Conjecture, 1979) Every planar graph G satisfies χe(G) = max{∆, w(G)}. The cases r = 0, 1, 2 are trivial. Seymour proved the exact conjecture for series-parrallel graphs in 1990.
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Seymour’s Exact Conjecture
For planar graph, Seymour proposed a stronger conjecture.
Conjecture
(Seymour’s Exact Conjecture, 1979) Every planar graph G satisfies χe(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 K3,3 or K −
5 as a minor, 2001.
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Seymour’s Exact Conjecture
For planar graph, Seymour proposed a stronger conjecture.
Conjecture
(Seymour’s Exact Conjecture, 1979) Every planar graph G satisfies χe(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 K3,3 or K −
5 as a minor, 2001.
The case r = 3 is equivalent to the Four Color Theorem.
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Seymour’s Exact Conjecture
For planar graph, Seymour proposed a stronger conjecture.
Conjecture
(Seymour’s Exact Conjecture, 1979) Every planar graph G satisfies χe(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 K3,3 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).
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Seymour’s Exact Conjecture
For planar graph, Seymour proposed a stronger conjecture.
Conjecture
(Seymour’s Exact Conjecture, 1979) Every planar graph G satisfies χe(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 K3,3 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
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Seymour’s Exact Conjecture
For planar graph, Seymour proposed a stronger conjecture.
Conjecture
(Seymour’s Exact Conjecture, 1979) Every planar graph G satisfies χe(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 K3,3 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.
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Seymour’s Exact Conjecture
For planar graph, Seymour proposed a stronger conjecture.
Conjecture
(Seymour’s Exact Conjecture, 1979) Every planar graph G satisfies χe(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 K3,3 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.
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Jakobsen’s critical graph conjecture
Conjecture
(Jakobosen’s Critical Graph Conjecture, 1973) Let G be a critical graph, and let χe(G) > m m − 1∆(G) + m − 3 m − 1. for an odd integer m ≥ 3. Then |V (G)| ≤ m − 2.
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Jakobsen’s critical graph conjecture
Conjecture
(Jakobosen’s Critical Graph Conjecture, 1973) Let G be a critical graph, and let χe(G) > m m − 1∆(G) + m − 3 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.
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Jakobsen’s critical graph conjecture
Conjecture
(Jakobosen’s Critical Graph Conjecture, 1973) Let G be a critical graph, and let χe(G) > m m − 1∆(G) + m − 3 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.
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Jakobsen’s critical graph conjecture
Conjecture
(Jakobosen’s Critical Graph Conjecture, 1973) Let G be a critical graph, and let χe(G) > m m − 1∆(G) + m − 3 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.
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Fractional edge chromatic number
An edge coloring can be considered as an Integer Programming Problem.
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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.
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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 Me denote the set of all matchings containing e.
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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 Me denote the set of all matchings containing e. χe(G) = min
yM, subject to: (1)
M∈Me yM = 1 for each edge e ∈ E(G).
(2) yM ∈ {0, 1}
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Fractional chromatic index
The fractional edge chromatic number χ∗
e(G) is defined as:
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Fractional chromatic index
The fractional edge chromatic number χ∗
e(G) is defined as:
χ∗
e(G) = min
yM, subject to: (1)
M∈Me yM = 1 for each edge e ∈ E(G).
(2) 0 ≤ yM ≤ 1
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Fractional chromatic index
The fractional edge chromatic number χ∗
e(G) is defined as:
χ∗
e(G) = min
yM, subject to: (1)
M∈Me yM = 1 for each edge e ∈ E(G).
(2) 0 ≤ yM ≤ 1 With matching techniques one can compute the fractional edge chromatic number in polynomial time.
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Fractional chromatic index
The fractional edge chromatic number χ∗
e(G) is defined as:
χ∗
e(G) = min
yM, subject to: (1)
M∈Me yM = 1 for each edge e ∈ E(G).
(2) 0 ≤ yM ≤ 1 With matching techniques one can compute the fractional edge chromatic number in polynomial time. From Edmond’s matching polytope theorem, χ∗
e(G) = max{∆,
max
H⊆G,|V (H)|≥2
|E(H)| ⌊ 1
2|V (H)|⌋}.
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Fractional chromatic index
The fractional edge chromatic number χ∗
e(G) is defined as:
χ∗
e(G) = min
yM, subject to: (1)
M∈Me yM = 1 for each edge e ∈ E(G).
(2) 0 ≤ yM ≤ 1 With matching techniques one can compute the fractional edge chromatic number in polynomial time. From Edmond’s matching polytope theorem, χ∗
e(G) = max{∆,
max
H⊆G,|V (H)|≥2
|E(H)| ⌊ 1
2|V (H)|⌋}.
If χ∗
e(G) = ∆, then w(G) ≤ ∆
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Fractional chromatic index
The fractional edge chromatic number χ∗
e(G) is defined as:
χ∗
e(G) = min
yM, subject to: (1)
M∈Me yM = 1 for each edge e ∈ E(G).
(2) 0 ≤ yM ≤ 1 With matching techniques one can compute the fractional edge chromatic number in polynomial time. From Edmond’s matching polytope theorem, χ∗
e(G) = max{∆,
max
H⊆G,|V (H)|≥2
|E(H)| ⌊ 1
2|V (H)|⌋}.
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.
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Fractional chromatic index
The computation of the edge chromatic number χe(G) is NP-hard
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Fractional chromatic index
The computation of the edge chromatic number χe(G) is NP-hard The fractional edge chromatic number can be computed in polynomial time.
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Fractional chromatic index
The computation of the edge chromatic number χe(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.
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Fractional chromatic index
The computation of the edge chromatic number χe(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
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Fractional chromatic index
The computation of the edge chromatic number χe(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 χe(G) = ⌈χ∗
e(G)⌉
for every graph G with χe(G) ≥ ∆ + 2.
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Fractional chromatic index
The computation of the edge chromatic number χe(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 χe(G) = ⌈χ∗
e(G)⌉
for every graph G with χe(G) ≥ ∆ + 2. χe(G) ≤ χ∗
e(G) +
e(G)/2 (Schiede, Sanders and Steuer 9/2 ).
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Fractional chromatic index
The computation of the edge chromatic number χe(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 χe(G) = ⌈χ∗
e(G)⌉
for every graph G with χe(G) ≥ ∆ + 2. χe(G) ≤ χ∗
e(G) +
e(G)/2 (Schiede, Sanders and Steuer 9/2 ).
Kahn proved in 1996 that every graph G satisfies χe(G) ∼ χ∗
e(G) as
χ∗
e(G) → ∞.
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Progress toward Goldberg’s Conjecture
Goldberg’s conjecture has been verified for the following cases: (Goldberg, 1974; Anderson, 1977) χe(G) > 5
4∆ + 2 4.
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Progress toward Goldberg’s Conjecture
Goldberg’s conjecture has been verified for the following cases: (Goldberg, 1974; Anderson, 1977) χe(G) > 5
4∆ + 2 4.
( Anderson, 1977) χe(G) > 7
6∆ + 4 6.
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Progress toward Goldberg’s Conjecture
Goldberg’s conjecture has been verified for the following cases: (Goldberg, 1974; Anderson, 1977) χe(G) > 5
4∆ + 2 4.
( Anderson, 1977) χe(G) > 7
6∆ + 4 6.
(Goldberg, 1984) χe(G) > 9
8∆ + 6 8.
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Progress toward Goldberg’s Conjecture
Goldberg’s conjecture has been verified for the following cases: (Goldberg, 1974; Anderson, 1977) χe(G) > 5
4∆ + 2 4.
( Anderson, 1977) χe(G) > 7
6∆ + 4 6.
(Goldberg, 1984) χe(G) > 9
8∆ + 6 8.
(Nishizeki, Kashiwagi, 1990; Tashkinov, 2000) χe(G) > 11
10∆ + 8 10.
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Progress toward Goldberg’s Conjecture
Goldberg’s conjecture has been verified for the following cases: (Goldberg, 1974; Anderson, 1977) χe(G) > 5
4∆ + 2 4.
( Anderson, 1977) χe(G) > 7
6∆ + 4 6.
(Goldberg, 1984) χe(G) > 9
8∆ + 6 8.
(Nishizeki, Kashiwagi, 1990; Tashkinov, 2000) χe(G) > 11
10∆ + 8 10.
(Favrholdt, Stiebitz, Toft, 2006) χe(G) > 13
12∆ + 10 12.
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Progress toward Goldberg’s Conjecture
Goldberg’s conjecture has been verified for the following cases: (Goldberg, 1974; Anderson, 1977) χe(G) > 5
4∆ + 2 4.
( Anderson, 1977) χe(G) > 7
6∆ + 4 6.
(Goldberg, 1984) χe(G) > 9
8∆ + 6 8.
(Nishizeki, Kashiwagi, 1990; Tashkinov, 2000) χe(G) > 11
10∆ + 8 10.
(Favrholdt, Stiebitz, Toft, 2006) χe(G) > 13
12∆ + 10 12.
(Scheide, Stiebitz, 2009) χe(G) > 15
14∆ + 12 14.
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Progress toward Goldberg’s Conjecture
Goldberg’s conjecture has been verified for the following cases: (Goldberg, 1974; Anderson, 1977) χe(G) > 5
4∆ + 2 4.
( Anderson, 1977) χe(G) > 7
6∆ + 4 6.
(Goldberg, 1984) χe(G) > 9
8∆ + 6 8.
(Nishizeki, Kashiwagi, 1990; Tashkinov, 2000) χe(G) > 11
10∆ + 8 10.
(Favrholdt, Stiebitz, Toft, 2006) χe(G) > 13
12∆ + 10 12.
(Scheide, Stiebitz, 2009) χe(G) > 15
14∆ + 12 14.
(Chen, Yu, Zang, 2011; Schied, 2010) χe(G) > ∆(G) +
2 .
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Progress toward Goldberg’s Conjecture
Goldberg’s conjecture has been verified for the following cases: (Goldberg, 1974; Anderson, 1977) χe(G) > 5
4∆ + 2 4.
( Anderson, 1977) χe(G) > 7
6∆ + 4 6.
(Goldberg, 1984) χe(G) > 9
8∆ + 6 8.
(Nishizeki, Kashiwagi, 1990; Tashkinov, 2000) χe(G) > 11
10∆ + 8 10.
(Favrholdt, Stiebitz, Toft, 2006) χe(G) > 13
12∆ + 10 12.
(Scheide, Stiebitz, 2009) χe(G) > 15
14∆ + 12 14.
(Chen, Yu, Zang, 2011; Schied, 2010) χe(G) > ∆(G) +
2 .
(Kurt, 2009) χe(G) > ∆(G) +
3
2 .
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Vizing’s Four Conjectures
In later 1960s, Vizing proposed the following four conjectures for SIMPLE graphs. (Vizing’s Independence Number Conjecture) The independence number of a critical graph is at most half of the number of vertices.
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Vizing’s Four Conjectures
In later 1960s, Vizing proposed the following four conjectures for SIMPLE graphs. (Vizing’s Independence Number Conjecture) The independence number of a critical graph is at most half of the number of vertices. (Vizing’s 2-factor Conjecture) Every critical graph has a 2-factor.
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Vizing’s Four Conjectures
In later 1960s, Vizing proposed the following four conjectures for SIMPLE graphs. (Vizing’s Independence Number Conjecture) The independence number of a critical graph is at most half of the number of vertices. (Vizing’s 2-factor Conjecture) Every critical graph has a 2-factor. (Vizing’s Conjecture on the Size of Critical Graphs) The average degree of a critical graph is at least ∆ − 1 + 3
n.
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Vizing’s Four Conjectures
In later 1960s, Vizing proposed the following four conjectures for SIMPLE graphs. (Vizing’s Independence Number Conjecture) The independence number of a critical graph is at most half of the number of vertices. (Vizing’s 2-factor Conjecture) Every critical graph has a 2-factor. (Vizing’s Conjecture on the Size of Critical Graphs) The average degree of a critical graph is at least ∆ − 1 + 3
n.
(Vizing’s Planar Graph Conjecture) Every planar graph with maximum degree 6 or 7 is class one.
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Other Conjectures
The Berge-Fulkerson Conjecture Every bridge less cubic graph G contains a family of six perfect matchings such that each edge of G is contained in precisely two of the matchings.
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Other Conjectures
The Berge-Fulkerson Conjecture Every bridge less cubic graph G contains a family of six perfect matchings such that each edge of G is contained in precisely two of the matchings. (Equivalence) For each bridge less cubic graph G, χe(2G) = 6.
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Other Conjectures
The Berge-Fulkerson Conjecture Every bridge less cubic graph G contains a family of six perfect matchings such that each edge of G is contained in precisely two of the matchings. (Equivalence) For each bridge less cubic graph G, χe(2G) = 6. (Berge’s conjecture) Every bridgeless cubic graph G contains a family
least one of the matchings.
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Other Conjectures
The Berge-Fulkerson Conjecture Every bridge less cubic graph G contains a family of six perfect matchings such that each edge of G is contained in precisely two of the matchings. (Equivalence) For each bridge less cubic graph G, χe(2G) = 6. (Berge’s conjecture) Every bridgeless cubic graph G contains a family
least one of the matchings. Mazzuoccolo proved that Berge’s conjecture is equivalent to The Berge-Fulkerson Conjecture.
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Other Conjectures
The Berge-Fulkerson Conjecture Every bridge less cubic graph G contains a family of six perfect matchings such that each edge of G is contained in precisely two of the matchings. (Equivalence) For each bridge less cubic graph G, χe(2G) = 6. (Berge’s conjecture) Every bridgeless cubic graph G contains a family
least one of the matchings. Mazzuoccolo proved that Berge’s conjecture is equivalent to The Berge-Fulkerson Conjecture. Matching cover: mk(G) = max{∪k
i=1Mi|
|E(G)| |M1, M2, · · · , Mkare perfect matchings ofG}.
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Other Conjectures
The Berge-Fulkerson Conjecture Every bridge less cubic graph G contains a family of six perfect matchings such that each edge of G is contained in precisely two of the matchings. (Equivalence) For each bridge less cubic graph G, χe(2G) = 6. (Berge’s conjecture) Every bridgeless cubic graph G contains a family
least one of the matchings. Mazzuoccolo proved that Berge’s conjecture is equivalent to The Berge-Fulkerson Conjecture. Matching cover: mk(G) = max{∪k
i=1Mi|
|E(G)| |M1, M2, · · · , Mkare perfect matchings ofG}.
m1(G) = 1
3 and Berge’s conjecture suggests that m5(G) = 1.
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Other conjectures see Chapter 9 of the book: Twenty Pretty Edge Coloring Conjectures
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Thank you!
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