Explicit Boij–Söderberg theory of ideals from a graph isomorphism reduction

Abstract

In the origins of complexity theory Booth and Lueker showed that the question of whether two graphs are isomorphic or not can be reduced to the special case of chordal graphs. To prove that, they defined a transformation from graphs G to chordal graphs BL(G). The projective resolutions of the associated edge ideals IBL(G) are manageable and we investigate to what extent their Betti tables also tell non-isomorphic graphs apart. It turns out that the coefficients describing the decompositions of Betti tables into pure diagrams in Boij–Söderberg theory are much more explicit than the Betti tables themselves, and they are expressed in terms of classical statistics of the graph G.