department of mathematics west virginia university · west virginia university ... introduction a...

Some Pretty Edge Coloring Conjectures

Rong Luo

Department of MathematicsWest Virginia University

Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 1 / 1

Introduction

A simple graph is a graph without multiple edges while a graphmeans parallel edges allowed.

An edge coloring of a graph is a function assigning values (colors) tothe edges of the graph in such a way that any two adjacent edgesreceive different colors.

The central problem in this area is to determine the minimum numberof colors needed for an edge coloring.

This is called the edge chromatic number, denoted χe = χe(G ).

χe ≥ ∆.

How high can edge chromatic number be?

Introduction

χe ≥ ∆.

Introduction

χe ≥ ∆.

Introduction

χe ≥ ∆.

Introduction

χe ≥ ∆.

Introduction

χe ≥ ∆.

History

Edge coloring first appeared in graph theory literature in the 1880’s.

In 1880, Tait was attempting to prove the Four Color Theorem, andhad shown

Theorem

(Tait) The Four Color Theorem is equivalent to that every 2-edgeconnected 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.

History

Theorem

History

Theorem

History

Theorem

History

Theorem

History

Theorem

History

Chapter 9, Twenty Pretty Edge Coloring Conjectures

Sample

9 (late 1960s), 6 (1970s), 4 (early 1980s), 1 (1990)

History

Chapter 9, Twenty Pretty Edge Coloring Conjectures

Sample

9 (late 1960s), 6 (1970s), 4 (early 1980s), 1 (1990)

History

Tashkinov, 2000, introduced a method, called Tashkinov tree, whichis 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)

History

Edge chromatic number

χe(G ) ≥ ∆.

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}.

χe(G ) ≥ ∆.

Theorem

χe(G ) ≥ ∆.

Theorem

χe(G ) ≥ ∆.

Theorem

χe(G ) ≥ ∆.

Theorem

χe(G ) ≥ ∆.

Theorem

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 ) ≤ ∆ + d∆−2go−1e (Goldberg, 1970s).

χe(G ) ≤ ∆ + d µb g

2ce (Stephen, 2001).

In general χe(G ) ∈ {∆,∆ + 1, · · · ,∆ + µ}.For a simple graph G , χe(G ) ∈ {∆,∆ + 1}.

χe(G ) ≤ 3∆2 (Shannon, 1949).

χe(G ) ≤ ∆ + d µb g

χe(G ) ≤ 3∆2 (Shannon, 1949).

χe(G ) ≤ ∆ + d µb g

χe(G ) ≤ 3∆2 (Shannon, 1949).

χe(G ) ≤ ∆ + d µb g

χe(G ) ≤ 3∆2 (Shannon, 1949).

χe(G ) ≤ ∆ + d µb g

In general χe(G ) ∈ {∆,∆ + 1, · · · ,∆ + µ}.

χe(G ) ≤ 3∆2 (Shannon, 1949).

χe(G ) ≤ ∆ + d µb g

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 | ≤ b |V (G)|2 c.

The total number of edges in G , |E (G )| ≤ kb |V (G)|2 c.

χe(G ) = k ≥ d |E(G)|b |V (G)|

For any subgraph H of G , we have χe(G ) ≥ χe(H) ≥ d |E(H)|b |V (H)|

χe(G ) ≥ maxH⊆G ,|V (H)|≥2

d |E (H)|b |V (H)|

where k = χe(G )

χe(G ) = k ≥ d |E(G)|b |V (G)|

χe(G ) ≥ maxH⊆G ,|V (H)|≥2

d |E (H)|b |V (H)|

where k = χe(G )

χe(G ) = k ≥ d |E(G)|b |V (G)|

χe(G ) ≥ maxH⊆G ,|V (H)|≥2

d |E (H)|b |V (H)|

where k = χe(G )

χe(G ) = k ≥ d |E(G)|b |V (G)|

χe(G ) ≥ maxH⊆G ,|V (H)|≥2

d |E (H)|b |V (H)|

where k = χe(G )

χe(G ) = k ≥ d |E(G)|b |V (G)|

χe(G ) ≥ maxH⊆G ,|V (H)|≥2

d |E (H)|b |V (H)|

where k = χe(G )

χe(G ) = k ≥ d |E(G)|b |V (G)|

χe(G ) ≥ maxH⊆G ,|V (H)|≥2

d |E (H)|b |V (H)|

χe(G ) ≥ maxH⊆G ,|V (H)|≥2

d |E (H)|b |V (H)

w(G ) = maxH⊆G ,|V (H)|≥2

d |E (H)|b |V (H)|

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 ) = maxH⊆G ,|V (H)|≥2

d |E (H)|b |V (H)|

2 ce = max

H⊆G ,|V (H)|≥3,oddd 2|E (H)||V (H)| − 1

χe(G ) ≥ maxH⊆G ,|V (H)|≥2

d |E (H)|b |V (H)

w(G ) = maxH⊆G ,|V (H)|≥2

d |E (H)|b |V (H)|

χe(G ) ≥ w(G )

w(G ) = maxH⊆G ,|V (H)|≥2

d |E (H)|b |V (H)|

2 ce = max

H⊆G ,|V (H)|≥3,oddd 2|E (H)||V (H)| − 1

χe(G ) ≥ maxH⊆G ,|V (H)|≥2

d |E (H)|b |V (H)

w(G ) = maxH⊆G ,|V (H)|≥2

d |E (H)|b |V (H)|

χe(G ) ≥ w(G )

w(G ) = maxH⊆G ,|V (H)|≥2

d |E (H)|b |V (H)|

2 ce = max

H⊆G ,|V (H)|≥3,oddd 2|E (H)||V (H)| − 1

χe(G ) ≥ maxH⊆G ,|V (H)|≥2

d |E (H)|b |V (H)

w(G ) = maxH⊆G ,|V (H)|≥2

d |E (H)|b |V (H)|

χe(G ) ≥ w(G )

w(G ) = maxH⊆G ,|V (H)|≥2

d |E (H)|b |V (H)|

2 ce = max

H⊆G ,|V (H)|≥3,oddd 2|E (H)||V (H)| − 1

χe(G ) ≥ maxH⊆G ,|V (H)|≥2

d |E (H)|b |V (H)

w(G ) = maxH⊆G ,|V (H)|≥2

d |E (H)|b |V (H)|

χe(G ) ≥ w(G )

χe(G ) ≥ max{∆,w(G )}.

The maximum value can be achieved when |V (H)| is odd.

w(G ) = maxH⊆G ,|V (H)|≥2

d |E (H)|b |V (H)|

2 ce = max

H⊆G ,|V (H)|≥3,oddd 2|E (H)||V (H)| − 1

χe(G ) ≥ maxH⊆G ,|V (H)|≥2

d |E (H)|b |V (H)

w(G ) = maxH⊆G ,|V (H)|≥2

d |E (H)|b |V (H)|

χe(G ) ≥ w(G )

w(G ) = maxH⊆G ,|V (H)|≥2

d |E (H)|b |V (H)|

2 ce = max

H⊆G ,|V (H)|≥3,oddd 2|E (H)||V (H)| − 1

χe(G ) ≥ maxH⊆G ,|V (H)|≥2

d |E (H)|b |V (H)

w(G ) = maxH⊆G ,|V (H)|≥2

d |E (H)|b |V (H)|

χe(G ) ≥ w(G )

w(G ) = maxH⊆G ,|V (H)|≥2

d |E (H)|b |V (H)|

2 ce = max

H⊆G ,|V (H)|≥3,oddd 2|E (H)||V (H)| − 1

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.

In general, χe(G ) ≥ max{∆,w(G )}.

Is it true χe(G ) ≤ max{∆,w(G )}+ 1.?

Conjecture

χe(G ) ≤ max{∆,w(G )}+ 1.

Conjecture

χe(G ) ≤ max{∆,w(G )}+ 1.

Conjecture

χe(G ) ≤ max{∆,w(G )}+ 1.

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 andindependently 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 )}.

Goldberg Conjecture

Conjecture

χe(G ) ≤ max{∆ + 1,w(G )}.

χe(G ) ∈ {∆,∆ + 1,w(G )}.

Goldberg Conjecture

Conjecture

χe(G ) ≤ max{∆ + 1,w(G )}.

χe(G ) ∈ {∆,∆ + 1,w(G )}.

Goldberg Conjecture

Conjecture

χe(G ) ≤ max{∆ + 1,w(G )}.

χe(G ) ∈ {∆,∆ + 1,w(G )}.

Goldberg Conjecture

Conjecture

χe(G ) ≤ max{∆ + 1,w(G )}.

χe(G ) ∈ {∆,∆ + 1,w(G )}.

Goldberg Conjecture–Importance

Goldberg Conjecture implies that if χe(G ) ≥ ∆ + 2, then there is apolynomial algorithm to compute χe(G ).

So it implies that the difficulty in determining χe(G ) is only todistinguish between two cases χe(G ) = ∆ and χe(G ) = ∆ + 1, whichis NP-hard proved by Holyer in 1980.

Goldberg Conjecture also implies Seymour’s r -graph conjecture andJakobsen’s critical graph conjecture.

G is critical if χe(G − e) < χe(G ) for any edge e.

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 satisfiesχe(G ) ≤ r + 1.

w(G ) ≤ r for each r -graph.Let H ⊆ G with |V (H)| odd. Let X = V (H). Then2|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 anr -graph. (Why?)Seymour proved that every graph with max{∆,w(G )} ≤ r iscontained in an r -graph as a subgraph.Since w(G ) ≤ r for each r -graph, Goldberg Conjecture impliesχe(G ) ≤ max{∆ + 1,w(G )} = r + 1.

Conjecture

2|E(H)||V (H)|−1 ≤

r(|X |−1)|X |−1 = r .

Conjecture

w(G ) ≤ r for each r -graph.

Let H ⊆ G with |V (H)| odd. Let X = V (H). Then2|E (H)| ≤ r |X | − r = r(|X | − 1)

2|E(H)||V (H)|−1 ≤

r(|X |−1)|X |−1 = r .

Conjecture

2|E(H)||V (H)|−1 ≤

r(|X |−1)|X |−1 = r .

Conjecture

2|E(H)||V (H)|−1 ≤

r(|X |−1)|X |−1 = r .

Conjecture

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 anr -graph. (Why?)

Seymour proved that every graph with max{∆,w(G )} ≤ r iscontained in an r -graph as a subgraph.Since w(G ) ≤ r for each r -graph, Goldberg Conjecture impliesχe(G ) ≤ max{∆ + 1,w(G )} = r + 1.

Conjecture

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 anr -graph. (Why?)Seymour proved that every graph with max{∆,w(G )} ≤ r iscontained in an r -graph as a subgraph.

Since w(G ) ≤ r for each r -graph, Goldberg Conjecture impliesχe(G ) ≤ max{∆ + 1,w(G )} = r + 1.

Conjecture

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 anr -graph. (Why?)Seymour proved that every graph with max{∆,w(G )} ≤ r iscontained in an r -graph as a subgraph.Since w(G ) ≤ r for each r -graph, Goldberg Conjecture impliesχe(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

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 )} orχe(tG ) = max{∆(tG ),w(tG )}+ 1.

Seymour’s Exact Conjecture

For planar graph, Seymour proposed a stronger conjecture.

Conjecture

(Seymour’s Exact Conjecture, 1979) Every planar graph G satisfiesχe(G ) = max{∆,w(G )}.

The cases r = 0, 1, 2 are trivial.

Seymour proved the exact conjecture for series-parrallel graphs in1990.

Marcotte verified the conjecture for the class of graphs not containingK3,3 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.

Conjecture

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’sconjecture.

Jakobsen’s conjecture was verified for 5 ≤ m ≤ 15.

Conjecture

χe(G ) >m

m − 1∆(G ) +

m − 3

m − 1.

Conjecture

χe(G ) >m

m − 1∆(G ) +

m − 3

m − 1.

Conjecture

χe(G ) >m

m − 1∆(G ) +

m − 3

m − 1.

Fractional edge chromatic number

An edge coloring can be considered as an Integer ProgrammingProblem.

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∑

M∈MyM ,

subject to:(1)

∑M∈Me

yM = 1 for each edge e ∈ E (G ).(2) yM ∈ {0, 1}

χe(G ) = min∑

M∈MyM ,

subject to:(1)

∑M∈Me

χe(G ) = min∑

M∈MyM ,

subject to:(1)

∑M∈Me

χe(G ) = min∑

M∈MyM ,

subject to:(1)

∑M∈Me

Fractional chromatic index

The fractional edge chromatic number χ∗e(G ) is defined as:

χ∗e(G ) = min∑

M∈MyM ,

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 edgechromatic number in polynomial time.

From Edmond’s matching polytope theorem,

χ∗e(G ) = max{∆, maxH⊆G ,|V (H)|≥2

|E (H)|b1

2 |V (H)|c}.

If χ∗e(G ) = ∆, then w(G ) ≤ ∆

If χ∗e(G ) > ∆, then w(G ) = dχ∗e(G )e and w(G ) can be computed inpolynomial time.

χ∗e(G ) = min∑

M∈MyM ,

subject to:(1)

∑M∈Me

χ∗e(G ) = max{∆, maxH⊆G ,|V (H)|≥2

|E (H)|b1

2 |V (H)|c}.

If χ∗e(G ) = ∆, then w(G ) ≤ ∆

χ∗e(G ) = min∑

M∈MyM ,

subject to:(1)

∑M∈Me

χ∗e(G ) = max{∆, maxH⊆G ,|V (H)|≥2

|E (H)|b1

2 |V (H)|c}.

If χ∗e(G ) = ∆, then w(G ) ≤ ∆

χ∗e(G ) = min∑

M∈MyM ,

subject to:(1)

∑M∈Me

χ∗e(G ) = max{∆, maxH⊆G ,|V (H)|≥2

|E (H)|b1

2 |V (H)|c}.

If χ∗e(G ) = ∆, then w(G ) ≤ ∆

χ∗e(G ) = min∑

M∈MyM ,

subject to:(1)

∑M∈Me

χ∗e(G ) = max{∆, maxH⊆G ,|V (H)|≥2

|E (H)|b1

2 |V (H)|c}.

If χ∗e(G ) = ∆, then w(G ) ≤ ∆

χ∗e(G ) = min∑

M∈MyM ,

subject to:(1)

∑M∈Me

χ∗e(G ) = max{∆, maxH⊆G ,|V (H)|≥2

|E (H)|b1

2 |V (H)|c}.

If χ∗e(G ) = ∆, then w(G ) ≤ ∆

The computation of the edge chromatic number χe(G ) is NP-hard

The fractional edge chromatic number can be computed inpolynomial time.

It is not clear whether the density w(G ) can be computed inpolynomial time.

max{∆(G ),w(G )} can be computed in polynomial time

Goldberg Conjecture is equivalent to the claim that χe(G ) = dχ∗e(G )efor every graph G with χe(G ) ≥ ∆ + 2.

χe(G ) ≤ χ∗e(G ) +√χ∗e(G )/2 (Schiede, Sanders and Steuer 9/2 ).

Kahn proved in 1996 that every graph G satisfies χe(G ) ∼ χ∗e(G ) asχ∗e(G )→∞.

Progress toward Goldberg’s Conjecture

Goldberg’s conjecture has been verified for the following cases:

(Goldberg, 1974; Anderson, 1977) χe(G ) > 54 ∆ + 2

( Anderson, 1977) χe(G ) > 76 ∆ + 4

(Goldberg, 1984) χe(G ) > 98 ∆ + 6

(Nishizeki, Kashiwagi, 1990; Tashkinov, 2000) χe(G ) > 1110 ∆ + 8

(Favrholdt, Stiebitz, Toft, 2006) χe(G ) > 1312 ∆ + 10

(Scheide, Stiebitz, 2009) χe(G ) > 1514 ∆ + 12

(Chen, Yu, Zang, 2011; Schied, 2010) χe(G ) > ∆(G ) +√

∆2 .

(Kurt, 2009) χe(G ) > ∆(G ) + 3

√∆2 .

( Anderson, 1977) χe(G ) > 76 ∆ + 4

(Goldberg, 1984) χe(G ) > 98 ∆ + 6

∆2 .

(Kurt, 2009) χe(G ) > ∆(G ) + 3

√∆2 .

( Anderson, 1977) χe(G ) > 76 ∆ + 4

(Goldberg, 1984) χe(G ) > 98 ∆ + 6

∆2 .

(Kurt, 2009) χe(G ) > ∆(G ) + 3

√∆2 .

( Anderson, 1977) χe(G ) > 76 ∆ + 4

(Goldberg, 1984) χe(G ) > 98 ∆ + 6

∆2 .

(Kurt, 2009) χe(G ) > ∆(G ) + 3

√∆2 .

( Anderson, 1977) χe(G ) > 76 ∆ + 4

(Goldberg, 1984) χe(G ) > 98 ∆ + 6

∆2 .

(Kurt, 2009) χe(G ) > ∆(G ) + 3

√∆2 .

( Anderson, 1977) χe(G ) > 76 ∆ + 4

(Goldberg, 1984) χe(G ) > 98 ∆ + 6

∆2 .

(Kurt, 2009) χe(G ) > ∆(G ) + 3

√∆2 .

( Anderson, 1977) χe(G ) > 76 ∆ + 4

(Goldberg, 1984) χe(G ) > 98 ∆ + 6

∆2 .

(Kurt, 2009) χe(G ) > ∆(G ) + 3

√∆2 .

( Anderson, 1977) χe(G ) > 76 ∆ + 4

(Goldberg, 1984) χe(G ) > 98 ∆ + 6

∆2 .

(Kurt, 2009) χe(G ) > ∆(G ) + 3

√∆2 .

Vizing’s Four Conjectures

In later 1960s, Vizing proposed the following four conjectures for SIMPLEgraphs.

(Vizing’s Independence Number Conjecture) The independencenumber 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 averagedegree of a critical graph is at least ∆− 1 + 3

(Vizing’s Planar Graph Conjecture) Every planar graph withmaximum degree 6 or 7 is class one.

Other Conjectures

The Berge-Fulkerson Conjecture Every bridge less cubic graph Gcontains a family of six perfect matchings such that each edge of G iscontained 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 familyof five perfect matchings such that each edge of G is contained in atleast one of the matchings.

Mazzuoccolo proved that Berge’s conjecture is equivalent to TheBerge-Fulkerson Conjecture.

Matching cover:

mk(G ) = max{∪ki=1Mi ||E(G)| |M1,M2, · · · ,Mkare perfect matchings ofG}.

m1(G ) = 13 and Berge’s conjecture suggests that m5(G ) = 1.

Other Conjectures

Matching cover:

Other Conjectures

Matching cover:

Other Conjectures

Matching cover:

Other Conjectures

Matching cover:

Other Conjectures

Matching cover:

Other conjectures see Chapter 9 of the book: Twenty Pretty EdgeColoring Conjectures

Sample

Thank you!

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