The Tutte Polynomial Modulo a Prime

For Fp the field on prime number p > 3 elements, it has been conjectured that there are just p+3 evaluations of the Tutte polynomial in Fp which are computable in polynomial time. In this note it is shown that if p 6≡ −1mod 12 then there are further polynomial-time computable evaluations. 1 Definitions and introduction Let G = (V,E) denote a graph, with loops and parallel edges permitted, and G the collection of all such graphs. The size of a graph G = (V,E) is |E|. If G has k(G) connected components, then the rank of G, denoted r(E), is |V | − k(G). The rank r(A) of a subset of edges A ⊆ E is the rank of the subgraph (V,A). Let X, Y be commuting indeterminates. The Tutte polynomial T (G;X, Y ) is a map T : G → Z[X, Y ] defined for all graphs G by T (G;X, Y ) = ∑ A⊆E (X − 1)r(E)−r(A)(Y − 1)|A|−r(A). (1) An evaluation of the Tutte polynomial in a commutative ring R with 1 is a map T (x, y) : G → Z[x, y] ⊆ R obtained from T by substituting (x, y) ∈ R×R for the indeterminate pair (X, Y ) in (1). For all commutative rings R with unity 1 and (x, y) ∈ R×R, (x− 1)(y − 1) = 1 ⇒ T (G;x, y) = x|E|(x− 1)r(E)−|E|. (2) 1 Partially supported by EPSRC Preprint submitted to Elsevier Preprint 15 August 2002 A theorem of [3] determines all the evaluations of the Tutte polynomial in C which are polynomial-time computable in the size of the graph G. Apart from those covered by (2) they are at the points (−1, 0), (0,−1), (1, 1), (−1,−1), (3) and (i,−i), (−i, i), (j, j), (j, j), (4) where i = √ −1 and j = (−1 + √ −3)/2. In [7, §7.5] it is shown that all evaluations of the Tutte polynomial in F2 are polynomial-time computable. All four evaluations in F2 reduce to finding the parity of evaluations in Z at points in (2) and (3). Annan [1] proved the following result. Theorem 1 [1, §3.6] Provided random polynomial time RP is not equal to NP, the only polynomial-time computable evaluations of the Tutte polynomial in F3 are at the points (−1, 0), (0,−1), (1, 1), (−1,−1) and (0, 0). He conjectured that similarly, for any prime p > 3, the only polynomialtime computable evaluations of the Tutte polynomial in Fp correspond to the points covered by (2) and the points (−1, 0), (0,−1), (1, 1), (−1,−1) in Fp×Fp corresponding to the points (3) in C× C. However, it will be shown that this conjecture needs to be modified to include further pairs of points corresponding to (4) when −1 or −3 is a square in Fp. 2 Polynomial-time evaluations of the Tutte polynomial in Fp Call a point (x, y) ∈ R×R easy if the evaluation T (G;x, y) in R is polynomialtime computable in the size of G. For a ring homomorphism π note that π(T (G;x, y)) = T (G; π(x), π(y)). The easy points (3) in Z × Z and the homomorphism π : Z → R, z 7→ z1, ensure that (−1, 0), (0,−1), (1, 1), (−1,−1) are easy in any commutative ring R with unity 1. This observation is made in [1, §3.6] for R = Fp. The following shows that the points of (4) yield further easy points in Fp for p ≡ 1mod 4 or p ≡ 1mod 3. The Legendre symbol (a/p) is defined to be +1 when a is a non-zero square in Fp and −1 when a is not a square.