The Theorem of R.L. BrooksTalk by Bjarne Toft at the Yoshimi Egawa 60 Conference.

Kagurazaka, Tokyo, Japan, September 10-14, 2013

Joint work with Michael Stiebitz.

Proc. Cambridge Phil. Soc. 1941

Colouring abstract graphs(rather than maps and graphs on surfaces)

MILESTONES:

• A.B.Kempe (1849-1922) 1879

• K. Wagner (1910-2000) 1937

• R.L. Brooks (1916- 1993) 1941

• H. Hadwiger (1908.1981) 1943

• G.A. Dirac (1925-1984) 1951

• T. Gallai (1912-1992) 1963

Klaus Wagner 1910-2000

Hugo Hadwiger 1908-1981

1973

Vierteljahrschr. der Naturf. Gesellschaft in Zürich 1943

It does not seem quite right to me that the conjecture now is named after me. In my Zürich lecture I just took your starting point, that contained the decisive idea, and carried it over to general chromatic numbers.

Rowland Leonard Brooks (1916 - 1993)

• Born February 6, 1916, in Lincolnshire, England

• Cambridge University 1935

• Tax-inspector in London

• Brooks’s famous note with his theorem was communicated to the Proc. Cambridge Phil. Soc. by W.T. Tutte in Nov. 1940 and published in 1941

• Died in London, June 18, 1993.

Further biographical information is not included

1993

Trinity Mathematical Society 1938

Trinity Four (Brooks, Smith, Stone, Tutte)

Smith and Brooks met at Trinity in 1935

• At the end of the first lecture Smith said to the young man sitting next to him: That was confusing.

• The young man answered: I thought it was a very good lecture. When is the next one?

• As for the next lecture the lecturer came in and started: IVANKEHIOSUTOKLSTMNDEJLSZIRTUNG

• The two realized after half an hour that they were perhaps in the wrongroom. They were! They left together, and the young man introducedhimself to Smith as Leonard Brooks.

• Leonard Brooks introduced Smith (and Stone) to a chess-playing friend (by the name Bill Tutte).

• A life-long friendship started!

Brooks’ Theorem 1941

• Let G be a graph of maximum degree Δ, where Δ ≠ 2, and supposethat that no connected component of G is a complete (Δ + 1)-graph. Then G has chromatic number at most Δ.

• Let G be a graph of maximum degree 2, and suppose that that noconnected component of G is an odd cycle. Then G has chromaticnumber 2.

ALTERNATIVELY:

• Let G be a connected graph of maximum degree Δ. Then G has chromatic number ≤ Δ+1, and it has chromatic number equal to Δ+1 if and only if it is a complete (Δ+1)-graph or an odd cycle.

Wiley 2014 ??

• Degree Bounds for the ChromaticNumber

• Degeneracy and Colorings

• Orientations and Colourings

• Properties of Critical Graphs

• Critical Graphs with Few Edges

• Homomorphisms and Colorings

• Coloring hypergraphs

• Coloring Graphs on Surfaces

• Graph Coloring Problems

Gabriel Andrew Dirac 1925-1984

CRITICAL GRAPHS WERE FIRST DEFINED IN G.A. DIRAC’s PhD-thesis 1951

Dirac’s Ph.D. Dissertation 1951

Dirac proved Brooks’ Theorem independently

Dénes König 1884-1944

Vielleicht noch mehr als der Berührung der Menschheit mit der Natur verdankt die Graphentheorie der Berührung der Menschen untereinander.

GALLAI made the picturesfor the book

Perhaps Graph Theory owes even more to the contact of human beings with human beings than to the contact of mankind with nature.

König in Göttingen 1904/05

Proofs of Brooks’ Theorem

• Sequential colouring and colour-interchange (Brooks 1942)

• Sequential colouring (Lovász 1975)

• Kempe chains (Melnikov and Vizing 1969)

• Maximum independent sets and Δ-reduction (Gerencsér 1965; Catlin1979; Tverberg 1983)

• Δ-Reduction (Rabern 2013)

• List-colourings (Erdős, Rubin and Taylor 1979)

• Critical graphs (Dirac 1951; Gallai 1963)

Sequential colouring

Δ-reduction (Landon Rabern - recent)

The case Δ = 3

List-colourings

• Let G be a connected graph, and let for each vertex v of G a list L(v) of at least d(v) different colours be given. Then G may be coloured suchthat each vertex gets a colour from its list, except if each block of G is either a complete graph or an odd cycle (G is a Gallai-tree).

In fact: except if

1) |L(v)| = d(v) for all v

2) If G has only one block, then L(v) is the same for all v

3) G is a Gallai-tree

Proof of the list colour theorem

• Suppose that (G,L) is a bad pair NOT SATISFYING 1), 2) and 3).

• If u is a non-separating vertex of G and c L(u), then let L’ denote the lists obtained from L by removing c from L(u) and from L(v) for all neighbours v of u in G.

• Then (G-u, L’) is a bad pair SATISFYING 1), 2) and 3)

We assume that (G,L) is a bad pair We shall prove that G satisfes 1), 2) and 3)

We know that G-u satisfies 1), 2) and 3)

Critical k-chromatic graphs

• BROOKS 1941: If G is k-critical (k≥4) on n vertices (n>k) then2𝑒 ≥ 𝑘 − 1 𝑛 + 1

• DIRAC 1957: If G is k-critical (k≥4) on n vertices (n>k) then2𝑒 ≥ 𝑘 − 1 𝑛 + (𝑘 − 3)

with equality for n= 2k-1.

• GALLAI 1963: 2𝑒 ≥ 𝑘 − 1 𝑛 + ((𝑘 − 3)/(𝑘2 − 3))𝑛

• KOSTOCHKA & STIEBITZ 1999: If G is k-critical (k≥4) on n vertices(n≥k+2 and n≠2k-1) then

2𝑒 ≥ 𝑘 − 1 𝑛 + 2(𝑘 − 3)

with equality for n=2k

We all have a favorite paper -I have two (both exactly 50 years old):

Or rather: I have three favorites!

• Gallai’s beautiful theory of alternating paths.

• The Gallai-Edmonds decomposition theorem.

• AN INTRIGUING FOOTNOTE :

• With the present methods I have succeeded in getting factorization theorems for general graphs besides σ=1 only for σ=2. I shall discussthese results on another occation

Smolenice June 1963

Tibor Gallai (1912 – 1992)

Critical graphs I• The blocks in the minor subgraphs are complete and/or odd

cycles (the minor graph is a forest of Gallai trees)

• This is best possible (by construction)

• If G is k-critical (k≥4) on n vertices (n>k) then2𝑒 ≥ 𝑘 − 1 𝑛 + ((𝑘 − 3)/(𝑘2 − 3))𝑛

• Gallai’s Conjecture:

2𝑒 ≥ [(𝑘2 − 𝑘 − 2)𝑛 − 𝑘(𝑘 − 3)]/(𝑘 − 1))

and this is sharp for n=1 mod (k-1)

• Krivelevich 1997, Kostochka&Stiebitz 1999, 2000 and 2002, Kostochka&Yancey 2012

Critical graphs II

• A k-critical graph with ≤ 2k-2 vertices has disconnected complement

• The proof uses Gallai’s theory of alternating paths from the 1950 paper

• Other proofs by Molloy 1999 and Stehlik 2003

• The right minimum number of edges for all n at most 2k-1

Hajós’ Construction

Ore’s Conjecture

f(k,n) = minimum number of edges in k-criticalgraph on n vertices (4 ≤ k ≤ n and n ≠ k+1)

• Dirac 1957: f(k,2k-1) known

• Gallai 1963: f(k, n) known for all n ≤ 2k-1

• Ore 1967: f(k,n+k-1) ≤ f(k,n) + k(k-1)/2 – 1.

• Kostochka and Stiebitz 1999: f(k,2k) is known.

• Equality in Ore’s inequality would therefore imply that f(k,n) is knownfor all values of k and n.

• Kostochka and Yancey 2013: f(k,n) is known for n = 1 mod (k-1)

• Fekete 1923: lim f(k,n)/n exists for all k, and it is known (2013).

• Kostochka and Yancey 2012: f(k,n) is known for k=4 and k=5.

f(4,n) = minimum number of edges in 4-critical graph on n vertices (4 ≤ n and n ≠ 5) • f(4,n) = the integer part of 5n/3, i.e.

• f(4,n) = 5n/3 for n = 0 mod 3

• f(4,n) = (5n-2)/3 for n = 1 mod 3

• f(4,n) = (5n-1)/3 for n = 2 mod 3

Proof outline of f(4,n) = 5n/3

Grötzsch’s Theorem (1959): Every planartriangle-free graph is 3-colourable.

4-critical graphs with empty minor graph

Planar 4-critical graphs without vertices of degree 3 (Koester 1984).

4-critical graphs with many edges/high min degree

4-critical graphs with all vertices of highdegree ( max min δ(G) )

• Simonovits and Toft 1971

• Max min d(G) ≥ c 3|𝑉 𝐺 |

• BEST POSSIBLE ??

Critical k-chromatic graphs (on n vertices) with just one Major vertex

• Min max |C| ≤ c log n

• BEST POSSIBLE –

• Alon, Krivelevich, Seymour 2000

• Shapira&Thomas 2011

• Max min |C| ~ c log n Erdős 1959 and 1962

• Max min |odd C| ??

Critical 4-chromatic graphs with long shortestodd cycles

• Max min |odd C| ≥ c 𝑛

• BEST POSSIBLE

Critical k-chromatic graphs with precisely twoMajor vertices• IF there are precisely two major

vertices and they areindependent

• THEN the minor graph is disconnected

• Gallai’s Conjecture: the numberof conn. components in the minor graph is at least the number of components in the major graph

• PROVED by Stiebitz in 1982

• USED by Krivelevich in 1997

A recent generalization by Landon Rabern

•Brooks’ Theorem: Every graph G with Δ(G) ≥ 3 satisfies (G) ≤ max{ , Δ}

• Landon Rabern 2013:Every graph G with Δ(G) ≥ 3 satisfies (G) ≤ max{ , Δ2 , 5(Δ+1)/6}, where Δ2 is the maximum degree of a vertex v adjacent to another vertex of degree at least as large as the degree of v.

Unsolved Problem 1

•Give the exact value of f(k,n) for all n ≥ k ≥ 4 and n ≠ k+1

•I.e. prove or disprove Ore’sConjecture that

f(k,n+k-1) = f(k,n) + k(k-1)/2 – 1

Unsolved Problem 2

•The exact value of f(4,n) for all n ≥ 4 and n ≠ 5 is known.

•Describe all the extremal 4-critical graphs

Unsolved Problem 3

•Determine the exact value of the minimum number of edges in a planar 4-critical graph on n vertices.

•Describe the extremal 4-critical graphs

All 4-critical graphs with at most 9 vertices

Unsolved Problem 4

•Prove or disprove the existenceof a constant c such that any 4-critical graph with minimum degreeδ satisfies |V(G)| ≥ cδ3.

Unsolved Problem 5

•Borodin & Kostochka 1977, Catlin 1978, Lawrence 1978 proved that(G) ≤ Δ(G) + 1 - (Δ+1)/(+1) provided3 ≤ (G) ≤ Δ(G).• It follows that (G) ≤ Δ(G) – 2 provided

3 ≤ (G) ≤ (Δ(G) – 2)/3 .• It follows that (G) ≤ Δ(G) – 3 provided

3 ≤ (G) ≤ (Δ(G) – 3)/4 .OBTAIN BETTER/OPTIMAL CONDITIONS

Results by Reed (1999), Cranston and Rabern(2013), Farzad, Molloy and Reed (2005)

• = Δ ≥ 1014 ⇒ = • = Δ ≥ 13 ⇒ ≥ -3• = Δ-1 large ⇒ ≥ -1 • = Δ-2 large ⇒ = -1• More general results by Molloy and Reed 2001 for large Δ

Unsolved Problem 6

•Borodin & Kostochka’s Conjecture1977:(G) ≤ Δ(G) – 1 provided(G) ≤ Δ(G) – 1 and Δ(G) ≥ 9.•I.e. : = Δ ≥ 9 ⇒ = •Proved for Δ(G) ≥ 1014 (Reed1999).

Unsolved Problem 7

•Reed 1999 conjectured:

•(G) ≤ 1

2(Δ(G) + 1) +

1

2(G)

•(G) ≤ 2

3(Δ(G) + 1) +

1

3(G) , provided Δ(G) ≥ 3

Unsolved Problem 8

•Brooks’ Theorem: If k ≥ 3 and G is a graph , not containing the complete bipartite graphK(1, k+1), nor the complete (k+1)-graph Kk+1, then (G) ≤ k.•Gyarfás’ Conjecture 1988: For any tree T thereis a function f such that if G is a graph not containing T as an induced subgraph then (G) ≤ f((G)).

Thank you for your attention.And congratulations, thanks and

all best wishes to Yoshimi Egawa!