A method in the study of real chromatic roots Thomas Perrett - - PowerPoint PPT Presentation

A method in the study of real chromatic roots

Thomas Perrett

Technical University of Denmark Joint work with Carsten Thomassen

16th June 2016

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The Chromatic Polynomial

The chromatic polynomial of a graph G is the unique univariate polynomial PG(q) such that for all q ∈ N, the number of q-colourings of G is precisely PG(q).

Figure : Chromatic polynomial of K2,3

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Chromatic Roots

Definition

A real number q is a chromatic root of a graph G if PG(q) = 0. Motivation for studying chromatic roots

1 Originally motivated by the Four Colour Theorem: ”4 is not a

chromatic root of a planar graph”.

2 Statistical physics and the Potts Model. 3 Mathematical interest. Thomas Perrett (DTU) 16th June 2016 3 / 21

Real chromatic roots

For a class of graphs, what can we say about the set of their chromatic roots?

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Real chromatic roots

For a class of graphs, what can we say about the set of their chromatic roots? What is its closure?

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Real chromatic roots

Theorem

For the class of all graphs:

• No chromatic roots in (−∞, 1] except 0 and 1. (Tutte ’74)
• No chromatic roots in (1, 32/27]. (Jackson ’93)
• Chromatic roots are dense in (32/27, ∞). (Thomassen ’97).

1 2 3 4 5

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Real chromatic roots

Theorem

For the class of all graphs:

• No chromatic roots in (−∞, 1] except 0 and 1. (Tutte ’74)
• No chromatic roots in (1, 32/27]. (Jackson ’93)
• Chromatic roots are dense in (32/27, ∞). (Thomassen ’97).

1 2 3 4 5

The closure of the set of chromatic roots of all graphs is {0, 1} ∪ [32/27, ∞).

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Real chromatic roots

Theorem

For the class of planar graphs:

• No chromatic roots in (−∞, 1] except 0 and 1. (Tutte ’74)
• No chromatic roots in (1, 32/27] (Jackson ’93)
• Chromatic roots are dense in (32/27, 3) (Thomassen ’97).
• No chromatic roots in [5, ∞) (Birkhoff & Lewis ’46).

Conjecture

For the class of planar graphs:

• Chromatic roots are dense in (3, 4) (Thomassen ’97).
• No chromatic roots in [4, 5] (Birkhoff & Lewis ’46).

1 2 3 4 5

? ?

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Thomassen’s Construction

Let A be a graph and e ∈ E(A).

e e e

t copies of C4 s copies of A

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Thomassen’s Construction

Let A be a graph and e ∈ E(A).

Thomas Perrett (DTU) 16th June 2016 7 / 21

Thomassen’s Construction

Let A be a graph and e ∈ E(A).

Thomas Perrett (DTU) 16th June 2016 7 / 21

Thomassen’s Construction

Let A be a graph and e ∈ E(A). If PA(q) and PA/e(q) satisfy a certain inequality at q ∈ R then for all ε > 0, there exists s, t ∈ N such that this graph has a chromatic root in (q − ε, q + ε).

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The condition

The condition we need is: (q − 1)PA/e(q) PA(q) < −1. For q ≥ 2, it suffices to find A such that

PA/e(q) PA(q) < −1.

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The condition

The condition we need is: (q − 1)PA/e(q) PA(q) < −1. For q ≥ 2, it suffices to find A such that

PA/e(q) PA(q) < −1.

Shortcut: Suppose PA(q) < 0 and that A is edge minimal with this property.

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The condition

The condition we need is: (q − 1)PA/e(q) PA(q) < −1. For q ≥ 2, it suffices to find A such that

PA/e(q) PA(q) < −1.

Shortcut: Suppose PA(q) < 0 and that A is edge minimal with this property. By deletion-contraction, PA(q) = PA−e(q) − PA/e(q). By minimality, PA−e(q) ≥ 0. But now PA/e(q) > 0 and PA/e(q) > |PA(q)|.

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Conclusion

Suppose G is a family of graphs such that G is a closed under edge deletion and making this construction. “For q ≥ 2, negativity in I implies density in I”

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Conclusion

Suppose G is a family of graphs such that G is a closed under edge deletion and making this construction. “For q ≥ 2, negativity in I implies density in I” Graph families that satisfy these conditions:

1 Planar graphs, 2 Any minor closed class where the excluded minors are 3-connected, 3 k-colourable graphs, etc... Thomas Perrett (DTU) 16th June 2016 9 / 21

Application 1: Planar graphs

Conjecture (Thomassen ’97)

The chromatic roots of planar graphs are dense in (3, 4).

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Application 1: Planar graphs

Conjecture (Thomassen ’97)

The chromatic roots of planar graphs are dense in (3, 4).

Theorem (P. & Thomassen ’16)

The chromatic roots of planar graphs are dense in (3, 3.61803) ∪ (3.61835, 4).

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Application 1: Planar graphs

Conjecture (Thomassen ’97)

The chromatic roots of planar graphs are dense in (3, 4).

Theorem (P. & Thomassen ’16)

The chromatic roots of planar graphs are dense in (3, 3.61803) ∪ (3.61835, 4). For each q in the set above, we just need to find planar graphs whose chromatic polynomials are negative at q.

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Planar graphs

Negative in (3, 3.61803 . . . ) Negative in (3.61835 . . . , 3.8). To get close to 4 we will need an infinite family of graphs.

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Close to 4

Royle’s analysis shows that, for any point q ∈ (3.7, 4), there exists a planar graph G such that PG(q) < 0.

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Mind the gap

Theorem (P. & Thomassen ’16)

The chromatic roots of planar graphs are dense in (3, 3.61803) ∪ (3.61835, 4).

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Mind the gap

Theorem (Tutte ’70)

The chromatic polynomial of a planar graph is positive at τ + 2, where τ ≈ 1.618034 is the golden ratio. Regarding the interval (τ + 2, 4), Read and Tutte ’88 wrote: “It is tempting to conjecture that the chromatic polynomial

• f a triangulation must be positive throughout this interval, but

counterexamples are known.”

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Application 2: Bipartite Graphs

Woodall showed that chromatic roots of G cannot be bounded in terms of the chromatic number χ(G).

Theorem (Woodall ’77)

For every q ∈ R \ N, there exist natural numbers n and m such that PKm,n(q) < 0. As a consequence, bipartite graphs can have arbitrarily large chromatic roots. Note that if G is bipartite then the graph constructed by Thomassen is also bipartite.

Corollary

The chromatic roots of bipartite graphs are dense in the interval [32/27, ∞).

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Open Problems 1: Mind the gap

Problem

Fill in the missing interval (3.61803, 3.61835). Because of Tutte’s Theorem you will need infinitely many graphs to use in Thomassen’s Construction. A variant of Royle’s construction could be useful.

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Open Problems 2: Other classes

There are interesting classes of graphs for which we cannot use this construction.

1 Planar triangulations. 2 Graphs embeddable on surfaces other than the plane. 3 3-connected graphs Thomas Perrett (DTU) 16th June 2016 17 / 21

Open Problems 2: Other classes: Planar triangulations

The Beraha numbers are defined by Bn = 2 + 2 cos 2π

n

• , n ≥ 1.

4, 0, 1, 2, τ + 1... For n ≥ 6 we have Bn ∈ [3, 4].

Conjecture (Beraha ’75)

There exists a planar triangulation with a real chromatic root in (Bn − ε, Bn + ε) for all n ≥ 1 and all ε > 0.

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Open Problems 2: Other classes: Surfaces

The only density results for graphs on the torus, say, come from what we know about planar graphs.

Question

Let S be a surface other than the plane. Is there an interval in (4, ∞) where the chromatic roots of graphs embeddable on S are dense?

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Open Problems 3: Other polynomials

The only results about density of flow roots come from those about chromatic roots via duality.

Theorem (Thomassen ’97, P. & Thomassen ’16)

The flow roots of graphs form a dense subset of [32/27, 3.61803 . . . ] ∪ [3.61835 . . . , 4].

Question

Does there exist a similar construction for flow polynomials?

Question

Do the flow roots form a dense subset of (4, 5)?

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Thanks.

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