Tutte polynomial

The Tutte polynomial factors into connected components. If ${\displaystyle G}$ is the union of disjoint graphs ${\displaystyle H}$ and ${\displaystyle H'}$ then

${\displaystyle T_{G}=T_{H}\cdot T_{H'}}$

If ${\displaystyle G}$ is planar and ${\displaystyle G^{*}}$ denotes its dual graph then

${\displaystyle T_{G}(x,y)=T_{G^{*}}(y,x)}$

Especially, the chromatic polynomial of a planar graph is the flow polynomial of its dual. Tutte refers to such functions as V-functions.[6]

Isomorphic graphs have the same Tutte polynomial, but the converse is not true. For example, the Tutte polynomial of every tree on ${\displaystyle m}$ edges is ${\displaystyle x^{m}}$.

Tutte polynomials are often given in tabular form by listing the coefficients ${\displaystyle t_{ij}}$ of ${\displaystyle x^{i}y^{j}}$ in row ${\displaystyle i}$ and column ${\displaystyle j}$. For example, the Tutte polynomial of the Petersen graph,

${\displaystyle {\begin{aligned}36x&+120x^{2}+180x^{3}+170x^{4}+114x^{5}+56x^{6}+21x^{7}+6x^{8}+x^{9}\\&+36y+84y^{2}+75y^{3}+35y^{4}+9y^{5}+y^{6}\\&+168xy+240x^{2}y+170x^{3}y+70x^{4}y+12x^{5}y\\&+171xy^{2}+105x^{2}y^{2}+30x^{3}y^{2}\\&+65xy^{3}+15x^{2}y^{3}\\&+10xy^{4},\end{aligned}}}$

is given by the following table.

Other example, the Tutte polynomial of the octahedron graph is given by

${\displaystyle {\begin{aligned}&12\,{y}^{2}{x}^{2}+11\,x+11\,y+40\,{y}^{3}+32\,{y}^{2}+46\,yx+24\,x{y}^{3}+52\,x{y}^{2}\\&+25\,{x}^{2}+29\,{y}^{4}+15\,{y}^{5}+5\,{y}^{6}+6\,{y}^{4}x\\&+39\,y{x}^{2}+20\,{x}^{3}+{y}^{7}+8\,y{x}^{3}+7\,{x}^{4}+{x}^{5}\end{aligned}}}$

The chromatic polynomial drawn in the Tutte plane

At ${\displaystyle y=0}$, the Tutte polynomial specialises to the chromatic polynomial,

${\displaystyle \chi _{G}(\lambda )=(-1)^{|V|-k(G)}\lambda ^{k(G)}T_{G}(1-\lambda ,0),}$

where ${\displaystyle k(G)}$ denotes the number of connected components of G.

For integer λ the value of chromatic polynomial ${\displaystyle \chi _{G}(\lambda )}$ equals the number of vertex colourings of G using a set of λ colours. It is clear that ${\displaystyle \chi _{G}(\lambda )}$ does not depend on the set of colours. What is less clear is that it is the evaluation at λ of a polynomial with integer coefficients. To see this, we observe:

1. If G has n vertices and no edges, then ${\displaystyle \chi _{G}(\lambda )=\lambda ^{n}}$.
2. If G contains a loop (a single edge connecting a vertex to itself), then ${\displaystyle \chi _{G}(\lambda )=0}$.
3. If e is an edge which is not a loop, then

${\displaystyle \chi _{G}(\lambda )=\chi _{G-e}(\lambda )-\chi _{G/e}(\lambda ).}$

The three conditions above enable us to calculate ${\displaystyle \chi _{G}(\lambda )}$, by applying a sequence of edge deletions and contractions; but they give no guarantee that a different sequence of deletions and contractions will lead to the same value. The guarantee comes from the fact that ${\displaystyle \chi _{G}(\lambda )}$ counts something, independently of the recurrence. In particular,

${\displaystyle T_{G}(2,0)=(-1)^{|V|}\chi _{G}(-1)}$

gives the number of acyclic orientations.

The Jones polynomial drawn in the Tutte plane

Along the hyperbola ${\displaystyle xy=1}$, the Tutte polynomial of a planar graph specialises to the Jones polynomial of an associated alternating knot.

${\displaystyle T_{G}(2,1)}$ counts the number of forests, i.e., the number of acyclic edge subsets.

${\displaystyle T_{G}(1,1)}$ counts the number of spanning forests (edge subsets without cycles and the same number of connected components as G). If the graph is connected, ${\displaystyle T_{G}(1,1)}$ counts the number of spanning trees.

${\displaystyle T_{G}(1,2)}$ counts the number of spanning subgraphs (edge subsets with the same number of connected components as G).

${\displaystyle T_{G}(2,0)}$ counts the number of acyclic orientations of G.[10]

${\displaystyle T_{G}(0,2)}$ counts the number of strongly connected orientations of G.[11]

${\displaystyle T_{G}(2,2)}$ is the number ${\displaystyle 2^{|E|}}$ where ${\displaystyle |E|}$ is the number of edges of graph G.

If G is a 4-regular graph, then

${\displaystyle (-1)^{|V|+k(G)}T_{G}(0,-2)}$

counts the number of Eulerian orientations of G. Here ${\displaystyle k(G)}$ is the number of connected components of G.[10]

If G is the m × n grid graph, then ${\displaystyle 2T_{G}(3,3)}$ counts the number of ways to tile a rectangle of width 4m and height 4n with T-tetrominoes.[12][13]

If G is a planar graph, then ${\displaystyle 2T_{G}(3,3)}$ equals the sum over weighted Eulerian orientations in a medial graph of G, where the weight of an orientation is 2 to the number of saddle vertices of the orientation (that is, the number of vertices with incident edges cyclicly ordered "in, out, in out").[14]

The partition functions for the Ising model and the 3- and 4-state Potts models drawn in the Tutte plane.

Define the hyperbola in the xy−plane:

${\displaystyle H_{2}:\quad (x-1)(y-1)=2,}$

the Tutte polynomial specialises to the partition function, ${\displaystyle Z(\cdot ),}$ of the Ising model studied in statistical physics. Specifically, along the hyperbola ${\displaystyle H_{2}}$ the two are related by the equation:[15]

${\displaystyle Z(G)=2\left(e^{-\alpha }\right)^{|E|-r(E)}\left(4\sinh \alpha \right)^{r(E)}T_{G}\left(\coth \alpha ,e^{2\alpha }\right).}$

In particular,

${\displaystyle (\coth \alpha -1)\left(e^{2\alpha }-1\right)=2}$

for all complex α.

More generally, if for any positive integer q, we define the hyperbola:

${\displaystyle H_{q}:\quad (x-1)(y-1)=q,}$

then the Tutte polynomial specialises to the partition function of the q-state Potts model. Various physical quantities analysed in the framework of the Potts model translate to specific parts of the ${\displaystyle H_{q}}$.

Correspondences between the Potts model and the Tutte plane [16]

Potts model	Tutte polynomial
Ferromagnetic	Positive branch of ${\displaystyle H_{q}}$
Antiferromagnetic	Negative branch of ${\displaystyle H_{q}}$ with ${\displaystyle y>0}$
High temperature	Asymptote of ${\displaystyle H_{q}}$ to ${\displaystyle y=1}$
Low temperature ferromagnetic	Positive branch of ${\displaystyle H_{q}}$ asymptotic to ${\displaystyle x=1}$
Zero temperature antiferromagnetic	Graph q-colouring, i.e., ${\displaystyle x=1-q,y=0}$

The flow polynomial drawn in the Tutte plane

At ${\displaystyle x=0}$, the Tutte polynomial specialises to the flow polynomial studied in combinatorics. For a connected and undirected graph G and integer k, a nowhere-zero k-flow is an assignment of “flow” values ${\displaystyle 1,2,\dots ,k-1}$ to the edges of an arbitrary orientation of G such that the total flow entering and leaving each vertex is congruent modulo k. The flow polynomial ${\displaystyle C_{G}(k)}$ denotes the number of nowhere-zero k-flows. This value is intimately connected with the chromatic polynomial, in fact, if G is a planar graph, the chromatic polynomial of G is equivalent to the flow polynomial of its dual graph ${\displaystyle G^{*}}$ in the sense that

Theorem (Tutte).

${\displaystyle C_{G}(k)=k^{-1}\chi _{G^{*}}(k).}$

The connection to the Tutte polynomial is given by:

${\displaystyle C_{G}(k)=(-1)^{|E|-|V|+k(G)}T_{G}(0,1-k).}$

The reliability polynomial drawn in the Tutte plane

At ${\displaystyle x=1}$, the Tutte polynomial specialises to the all-terminal reliability polynomial studied in network theory. For a connected graph G remove every edge with probability p; this models a network subject to random edge failures. Then the reliability polynomial is a function ${\displaystyle R_{G}(p)}$, a polynomial in p, that gives the probability that every pair of vertices in G remains connected after the edges fail. The connection to the Tutte polynomial is given by

${\displaystyle R_{G}(p)=(1-p)^{|V|-k(G)}p^{|E|-|V|+k(G)}T_{G}\left(1,{\tfrac {1}{p}}\right).}$

Tutte also defined a closer 2-variable generalization of the chromatic polynomial, the dichromatic polynomial of a graph. This is

${\displaystyle Q_{G}(u,v)=\sum \nolimits _{A\subseteq E}u^{k(A)}v^{|A|-|V|+k(A)},}$

where ${\displaystyle k(A)}$ is the number of connected components of the spanning subgraph (V,A). This is related to the corank-nullity polynomial by

${\displaystyle Q_{G}(u,v)=u^{k(G)}\,R_{G}(u,v).}$

The dichromatic polynomial does not generalize to matroids because k(A) is not a matroid property: different graphs with the same matroid can have different numbers of connected components.

The Martin polynomial ${\displaystyle m_{\vec {G}}(x)}$ of an oriented 4-regular graph ${\displaystyle {\vec {G}}}$ was defined by Pierre Martin in 1977.[17] He showed that if G is a plane graph and ${\displaystyle {\vec {G}}_{m}}$ is its directed medial graph, then

${\displaystyle T_{G}(x,x)=m_{{\vec {G}}_{m}}(x).}$

The deletion–contraction algorithm applied to the diamond graph. Red edges are deleted in the left child and contracted in the right child. The resulting polynomial is the sum of the monomials at the leaves, ${\displaystyle x^{3}+2x^{2}+y^{2}+2xy+x+y}$. Based on Welsh & Merino (2000).

The deletion–contraction recurrence for the Tutte polynomial,

${\displaystyle T_{G}(x,y)=T_{G\setminus e}(x,y)+T_{G/e}(x,y),\qquad e{\text{ not a loop nor a bridge.}}}$

immediately yields a recursive algorithm for computing it for a given graph: as long as you can find an edge e that is not a loop or bridge, recursively compute the Tutte polynomial for when that edge is deleted, and when that edge is contracted. Then add the two sub-results together to get the overall Tutte polynomial for the graph.

The base case is a monomial ${\displaystyle x^{m}y^{n}}$ where m is the number of bridges, and n is the number of loops.

Within a polynomial factor, the running time t of this algorithm can be expressed in terms of the number of vertices n and the number of edges m of the graph,

${\displaystyle t(n+m)=t(n+m-1)+t(n+m-2),}$

a recurrence relation that scales as the Fibonacci numbers with solution[18]

${\displaystyle t(n+m)=\left({\frac {1+{\sqrt {5}}}{2}}\right)^{n+m}=O\left(1.6180^{n+m}\right).}$

The analysis can be improved to within a polynomial factor of the number ${\displaystyle \tau (G)}$ of spanning trees of the input graph.[19] For sparse graphs with ${\displaystyle m=O(n)}$ this running time is ${\displaystyle \exp(O(n))}$. For regular graphs of degree k, the number of spanning trees can be bounded by

${\displaystyle \tau (G)=O\left(\nu _{k}^{n}n^{-1}\log n\right),}$

where

${\displaystyle \nu _{k}={\frac {(k-1)^{k-1}}{(k^{2}-2k)^{{\frac {k}{2}}-1}}}.}$

so the deletion–contraction algorithm runs within a polynomial factor of this bound. For example:[20]

${\displaystyle \nu _{5}\approx 4.4066.}$

In practice, graph isomorphism testing is used to avoid some recursive calls. This approach works well for graphs that are quite sparse and exhibit many symmetries; the performance of the algorithm depends on the heuristic used to pick the edge e.[19][21][22]

In some restricted instances, the Tutte polynomial can be computed in polynomial time, ultimately because Gaussian elimination efficiently computes the matrix operations determinant and Pfaffian. These algorithms are themselves important results from algebraic graph theory and statistical mechanics.

${\displaystyle T_{G}(1,1)}$ equals the number ${\displaystyle \tau (G)}$ of spanning trees of a connected graph. This is computable in polynomial time as the determinant of a maximal principal submatrix of the Laplacian matrix of G, an early result in algebraic graph theory known as Kirchhoff’s Matrix–Tree theorem. Likewise, the dimension of the bicycle space at ${\displaystyle T_{G}(-1,-1)}$ can be computed in polynomial time by Gaussian elimination.

For planar graphs, the partition function of the Ising model, i.e., the Tutte polynomial at the hyperbola ${\displaystyle H_{2}}$, can be expressed as a Pfaffian and computed efficiently via the FKT algorithm. This idea was developed by Fisher, Kasteleyn, and Temperley to compute the number of dimer covers of a planar lattice model.

Using a Markov chain Monte Carlo method, the Tutte polynomial can be arbitrarily well approximated along the positive branch of ${\displaystyle H_{2}}$, equivalently, the partition function of the ferromagnetic Ising model. This exploits the close connection between the Ising model and the problem of counting matchings in a graph. The idea behind this celebrated result of Jerrum and Sinclair[23] is to set up a Markov chain whose states are the matchings of the input graph. The transitions are defined by choosing edges at random and modifying the matching accordingly. The resulting Markov chain is rapidly mixing and leads to “sufficiently random” matchings, which can be used to recover the partition function using random sampling. The resulting algorithm is a fully polynomial-time randomized approximation scheme (fpras).

The Tutte plane. Every point ${\displaystyle (x,y)}$ in the real plane corresponds to a computational problem ${\displaystyle T_{G}(x,y)}$. At any red point, the problem is polynomial-time computable; at any blue point, the problem is #P-hard in general, but polynomial-time computable for planar graphs; and at any point in the white regions, the problem is #P-hard even for bipartite planar graphs.

If both x and y are non-negative integers, the problem ${\displaystyle T_{G}(x,y)}$ belongs to #P. For general integer pairs, the Tutte polynomial contains negative terms, which places the problem in the complexity class GapP, the closure of #P under subtraction. To accommodate rational coordinates ${\displaystyle (x,y)}$, one can define a rational analogue of #P.[24]

The computational complexity of exactly computing ${\displaystyle T_{G}(x,y)}$ falls into one of two classes for any ${\displaystyle x,y\in \mathbb {C} }$. The problem is #P-hard unless ${\displaystyle (x,y)}$ lies on the hyperbola ${\displaystyle H_{1}}$ or is one of the points

${\displaystyle \left\{(1,1),(-1,-1),(0,-1),(-1,0),(i,-i),(-i,i),\left(j,j^{2}\right),\left(j^{2},j\right)\right\},\qquad j=e^{\frac {2\pi i}{3}}.}$

in which cases it is computable in polynomial time.[25] If the problem is restricted to the class of planar graphs, the points on the hyperbola ${\displaystyle H_{2}}$ become polynomial-time computable as well. All other points remain #P-hard, even for bipartite planar graphs.[26] In his paper on the dichotomy for planar graphs, Vertigan claims (in his conclusion) that the same result holds when further restricted to graphs with vertex degree at most three, save for the point ${\displaystyle T_{G}(0,-2)}$, which counts nowhere-zero Z₃-flows and is computable in polynomial time.[27]

These results contain several notable special cases. For example, the problem of computing the partition function of the Ising model is #P-hard in general, even though celebrated algorithms of Onsager and Fisher solve it for planar lattices. Also, the Jones polynomial is #P-hard to compute. Finally, computing the number of four-colorings of a planar graph is #P-complete, even though the decision problem is trivial by the four color theorem. In contrast, it is easy to see that counting the number of three-colorings for planar graphs is #P-complete because the decision problem is known to be NP-complete via a parsimonious reduction.

The question which points admit a good approximation algorithm has been very well studied. Apart from the points that can be computed exactly in polynomial time, the only approximation algorithm known for ${\displaystyle T_{G}(x,y)}$ is Jerrum and Sinclair’s FPRAS, which works for points on the “Ising” hyperbola ${\displaystyle H_{2}}$ for y > 0. If the input graphs are restricted to dense instances, with degree ${\displaystyle \Omega (n)}$, there is an FPRAS if x ≥ 1, y ≥ 1.[28]

Even though the situation is not as well understood as for exact computation, large areas of the plane are known to be hard to approximate.[24]