Polynomial graph invariants from homomorphism numbers

The number of homomorphisms hom(G,Kk) from a graph G to the complete graph Kk is the value of the chromatic polynomial of G at a positive integer k. This motivates the following (cf. [3]): Definition 1 A sequence of graphs (Hk), k = (k1, . . . , kh) ∈ N , is strongly polynomial if for every graph G there is a polynomial p(G; x1, . . . , xh) such that hom(G,Hk) = p(G; k1, . . . , kh) for every k ∈ N . Many important graph polynomials p(G) are determined by strongly polynomial sequences of graphs (Hk): e.g. [2] the Tutte polynomial, Averbouch–Godlin– Makowsky polynomial [1] (includes the matching polynomial) and Tittmann– Averbouch–Godlin polynomial [4] (includes the independence polynomial). We give a new construction of strongly polynomial sequences based on coloured rooted tree encodings of graphs (such as cotrees for cographs), which among other things offers a natural generalization of the above polynomials. In this talk we illustrate this method with the following. We start with a simple graph H given as a spanning subgraph of the closure of a rooted tree T . For each k= (ks : s∈ V (T )) ∈ N |V (T )| we use the tree T to recursively construct a graph T (H), in which, for each s ∈ V (T ), we create ks isomorphic copies of the subtree Ts of T rooted at s, all pendant from the same vertex as Ts, while propagating adjacencies of H in the closure of T to these copies of Ts. Theorem 1 The sequence (T (H)) is strongly polynomial. Define β(H) to be the minimum value of |V (T )| such that H is a subgraph of the closure of T (T ). For example, β(K1,l) = 2, β(P2l) = 2l, β(P2l−1) = l, and β(Kl) = l. We have tree-depth td(H) ≤ β(H) and β(H) = |V (H)| if H has no involutive automorphisms. Theorem 2 Let H be a family of simple graphs such that {β(H) : H ∈ H} is bounded. Then H can be partitioned into a finite number of subsequences of strongly polynomial sequences of graphs.