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 I BL(G) is 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

Linearization of monomial ideals

“…In 1975 Booth and Lueker introduced a construction, in their paper [4], that takes an arbitrary finite simple graph G and returns a graph with more structure, which is denoted as BL(G) in [13] and defined as follows.…”

“…One can see that the complement of BL(G) is chordal, and therefore the edge ideal I BL(G) has a linear resolution. This and related matters are addressed in [13]. After that paper was made available, it was remarked that the Booth-Lueker construction could be interpreted as a map that associates to any squarefree monomial ideal generated in degree 2 a monomial ideal, also squarefree and generated in degree 2, with linear resolution.…”