A conjecture on independent sets and graph covers

An M -cover of a graph G is its M -fold covering space 1 . All M -covers are explicitly constructed using permutation voltage assignment as follows [1]. A permutation voltage assignment of G is a map

where S M is the permutation group of {1, . . . , M }. Then an M -cover G = ( Ṽ , Ẽ) of G is given by Ṽ := V × {1, . . . , M } and

If an M -cover is M copies of G then it is called trivial M -cover and denoted by G ⊕M . This is obtained by identity permutations. The natural projection, π, from a cover G to G is obtained by forgetting the “layer number”. That is, π : Ṽ → V is given by π(u, i) = u and π : Ẽ → E is given by π((v, k)(u, l)) = vu.

An independent set I of a graph G is a subset of V such that none of the elements in I are adjacent in G. Formally, I is an independent set iff u, v ∈ I ⇒ uv ∈ E. The multivariate independent set polynomial of G is defined by p(G) :=

with indeterminates x v (v ∈ V ).

URL: watay@ism.ac.jp (Yusuke Watanabe) 1 Interpret graphs as topological spaces.

November 23, 2018

We extend the definition of the projection map π over the multivariate polynomial ring. First, let us define a map Π for each indeterminate by Π(x v ) = x Π(v) , where v ∈ Ṽ . Then this is uniquely extended to the polynomial ring as a ring homomorphism; for example Π(

Conjecture 1. 2 For any bipartite graph G and its M -cover G, we conjecture the following relation:

where Π is defined as above and the symbol means the inequalities for all coefficients of monomials.

We can interpret the conjecture more explicitly as follows. For a subset U of V , define I( G, U ) := {I ⊂ Ṽ |I is independent set, π(I) = U }, where π(I) is the image of I. Since p(G) M = Π(p(G ⊕M )), the conjecture is equivalent to the following statement:

Example 1. Let G be a cycle graph of length four and let G be its 3-cover that is isomorphic to the cycle of length twelve. Then,

x (v,m) + . . . ,

Π(p( G)) = 1 + 3(x 1 + x 2 + x 3 + x 4 ) + . . . .

It takes time and effort to check the conjecture, however, it is true in this case.

Remark. The above conjecture is claimed for the pair (bipartite graph, independent set). I also conjecture analogous properties for (bipartite graph, matching), (graph with even number of vertices, perfect matching) and (graph, Eulerian set3 ).

The conjecture originates from the theory of the Bethe approximation. The partition function of a binary pairwise model on a graph G is

where the weights (J , h) are called interactions. 4 It is called attractive if J vu ≥ 0 for all vu ∈ E. The Bethe partition function5 Z B is defined by [2] Z B : = exp -min

where F B is the Bethe free energy and < • > is the mean with respect to the M ! |E| covers. (Details are omitted. See [2].)

holds for any binary pairwise attractive models.6

Proof. From (11), the assertion of the theorem is proved if we show that

for any M -cover G of G. In the following, we see that the partition function can be written by the independent set polynomial and thus the above inequality holds under the assumption of Conjecture 1.

where d v (S) is the number of edges in S connecting to v, A uv = e Juv -1, B v = e -hv and G ′ is a bipartite graph obtained by adding a new vertex on each edge of G.

Remark. The Bethe approximation can also be applied to the computation of the permanent of non-negative matrices. From the combinatorial viewpoint, this problem is related to the (weighted) perfect matching problem on complete bipartite graphs. Vontobel analyzed this problem and pose a conjecture analogous to Eq.( 5) [4]. This conjecture implies the inequality between the permanent and its Bethe approximation, Z ≥ Z B , given his formula Eq.( 11). The statement Z ≥ Z B is, however, directly proved by Gurvits, generalizing Schrijverfs permanental inequality [5].

I have checked the conjecture for many examples by computer.

A subset of edges is Eulerian if it induces a subgraph that only has vertices of degree two and zero.

In the following, for a cover G of G, we think that interactions are naturally induced from G.

This is computed from the absolute minimum of the Bethe free energy; other Bethe approximations of the partition function corresponding to local minima are smaller than Z B .

In a quite limited situation, the inequality is proved in[3].