JournalsaihpdVol. 1, No. 1pp. 47–60

The Potts model and chromatic functions of graphs

  • Martin Klazar

    Charles University, Prague, Czech Republic
  • Martin Loebl

    Charles University, Prague, Czech Republic
  • Iain Moffatt

    Royal Holloway, University of London, Egham, UK
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Abstract

The UU-polynomial of Noble and Welsh is known to have intimate connections with the Potts model as well as with several important graph polynomials. For each graph GG, U(G)U(G) is equivalent to the Stanley's symmetric bad colouring polynomial XB(G)XB(G). Moreover Sarmiento established the equivalence between UU and the polychromate of Brylawski. All these functions have countable number of variables, even though the restrictions to an arbitrary graph are honest polynomials. Loebl defined the qq-dichromate Bq(G,x,y)B_q(G,x,y) as a function of graph GG and three independent variables q,x,yq,x,y, proved that it is equal to the partition function of the Potts model with variable number of states and with certain magnetic field contribution, and conjectured that qq-dichromate is equivalent to the UU-polynomial. He also proposed a stronger conjecture on integer partitions. The aim of this paper is two-fold. We present a construction disproving the Loebl's integer partitions conjecture, and we introduce a new function Br,q(G,x,k)B_{r,q}(G,x,k) which is also equal to the partition function of the Potts model with variable number of states and with a (different) external field contribution, and we show that Br,q(G,x,k)B_{r,q}(G,x,k) is equivalent to UU-polynomial. This gives a Potts model-type formulation for the UU-polynomial.

Cite this article

Martin Klazar, Martin Loebl, Iain Moffatt, The Potts model and chromatic functions of graphs. Ann. Inst. Henri Poincaré Comb. Phys. Interact. 1 (2014), no. 1, pp. 47–60

DOI 10.4171/AIHPD/2