Torsion in the first group of the chromatic graph cohomology over algebras Z[x]/(x^m)
by
Radmila Sazdanovic
George Washington University
Coauthors: Jozef H. Przytycki, Milena Pabiniak

In this talk we focus on the first chromatic graph cohomology over algebra Z[x]/(x^m) of truncated polynomials. In particular, we are interested in grading motivated by the interpretation of Hochschild homology as graph cohomology of polygons and its generalization to arbitrary graphs. As an introduction to this talk, my coauthors will give the complete description of H^{1, v-1}_A2(G) and H^{1, 2v-3}_A3(G). However, for Z[x]/(x^m) and m > 3 we give only conjectures based both on theory developed for m=2, 3 and computational results.

Theorem 1.
For complete graph with n vertices K_n, n ≥ 4 have

H1, 2n-3A3(Kn) = Z2⊕Z3n-1⊕Z[(n(n-1)(2n-7))/6].

Corollary 2.
If a graph G contains a triangle then H^{1, 2v(G)-3}_A3(G) contains Z₃ torsion.

We show that (∀n) (∃ simple G) Z_n ∈  torH^{1, 2v-3}_A3(G).

Conjecture 3.
For any graph W_n with n vertices where one vertex is of degree n-1 and all the rest are of degree 3 (wheel), n > 4 and m ≥ 4 the following holds:

H1, 4n-3Am(Wnout)=Zmn-1 ⊕Zn-2

H1, 4n-3Am(Wn)=Zmn ⊕Zn

H1, 4n-3Am(Wnin)=Zmn-2⊕Zn-2

Conjecture 4.
For complete graph K_n where n ≥ 4 and odd m > 3 the following is true:

H1, (m-1)(n-2)+1Am(Kn) = Zm[(n (n-1)(n-2))/6] ⊕Z2[(n (n-1)(n-2)(n-3))/24] ⊕Z[(n (n-1)(n-2)(n-3))/12]

Moreover, we will present some computational results for width of H¹_A3(G) of several families of graphs graphs.

Date received: May 4, 2006