A method in the study of real chromatic roots Thomas Perrett Technical University of Denmark Joint work with Carsten Thomassen 16th June 2016 Thomas Perrett (DTU) 16th June 2016 1 / 21
The Chromatic Polynomial The chromatic polynomial of a graph G is the unique univariate polynomial P G ( q ) such that for all q ∈ N , the number of q -colourings of G is precisely P G ( q ). Figure : Chromatic polynomial of K 2 , 3 Thomas Perrett (DTU) 16th June 2016 2 / 21
Chromatic Roots Definition A real number q is a chromatic root of a graph G if P G ( 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? Thomas Perrett (DTU) 16th June 2016 4 / 21
Real chromatic roots For a class of graphs, what can we say about the set of their chromatic roots? What is its closure ? Thomas Perrett (DTU) 16th June 2016 4 / 21
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) . 0 1 2 3 4 5 Thomas Perrett (DTU) 16th June 2016 5 / 21
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) . 0 1 2 3 4 5 The closure of the set of chromatic roots of all graphs is { 0 , 1 } ∪ [32 / 27 , ∞ ). Thomas Perrett (DTU) 16th June 2016 5 / 21
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) . ? ? 0 1 2 3 4 5 Thomas Perrett (DTU) 16th June 2016 6 / 21
Thomassen’s Construction Let A be a graph and e ∈ E ( A ). e e e s copies of A t copies of C 4 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 ). Thomas Perrett (DTU) 16th June 2016 7 / 21
Thomassen’s Construction Let A be a graph and e ∈ E ( A ). If P A ( q ) and P A / 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 ). Thomas Perrett (DTU) 16th June 2016 7 / 21
The condition The condition we need is: ( q − 1) P A / e ( q ) < − 1 . P A ( q ) P A / e ( q ) For q ≥ 2, it suffices to find A such that P A ( q ) < − 1. Thomas Perrett (DTU) 16th June 2016 8 / 21
The condition The condition we need is: ( q − 1) P A / e ( q ) < − 1 . P A ( q ) P A / e ( q ) For q ≥ 2, it suffices to find A such that P A ( q ) < − 1. Shortcut: Suppose P A ( q ) < 0 and that A is edge minimal with this property. Thomas Perrett (DTU) 16th June 2016 8 / 21
The condition The condition we need is: ( q − 1) P A / e ( q ) < − 1 . P A ( q ) P A / e ( q ) For q ≥ 2, it suffices to find A such that P A ( q ) < − 1. Shortcut: Suppose P A ( q ) < 0 and that A is edge minimal with this property. By deletion-contraction, P A ( q ) = P A − e ( q ) − P A / e ( q ) . By minimality, P A − e ( q ) ≥ 0. But now P A / e ( q ) > 0 and P A / e ( q ) > | P A ( q ) | . Thomas Perrett (DTU) 16th June 2016 8 / 21
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 ” Thomas Perrett (DTU) 16th June 2016 9 / 21
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) . Thomas Perrett (DTU) 16th June 2016 10 / 21
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) . Thomas Perrett (DTU) 16th June 2016 10 / 21
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 . Thomas Perrett (DTU) 16th June 2016 10 / 21
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. Thomas Perrett (DTU) 16th June 2016 11 / 21
Close to 4 Royle’s analysis shows that, for any point q ∈ (3 . 7 , 4), there exists a planar graph G such that P G ( q ) < 0. Thomas Perrett (DTU) 16th June 2016 12 / 21
Mind the gap Theorem (P. & Thomassen ’16) The chromatic roots of planar graphs are dense in (3 , 3 . 61803) ∪ (3 . 61835 , 4) . Thomas Perrett (DTU) 16th June 2016 13 / 21
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 of a triangulation must be positive throughout this interval, but counterexamples are known.” Thomas Perrett (DTU) 16th June 2016 14 / 21
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 P K m , 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 , ∞ ) . Thomas Perrett (DTU) 16th June 2016 15 / 21
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. Thomas Perrett (DTU) 16th June 2016 16 / 21
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 � 2 π � The Beraha numbers are defined by B n = 2 + 2 cos , n ≥ 1. n 4, 0, 1, 2, τ + 1... For n ≥ 6 we have B n ∈ [3 , 4]. Conjecture (Beraha ’75) There exists a planar triangulation with a real chromatic root in ( B n − ε, B n + ε ) for all n ≥ 1 and all ε > 0 . Thomas Perrett (DTU) 16th June 2016 18 / 21
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? Thomas Perrett (DTU) 16th June 2016 19 / 21
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) ? Thomas Perrett (DTU) 16th June 2016 20 / 21
Thanks. Thomas Perrett (DTU) 16th June 2016 21 / 21
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