HOMOLOGICAL AND COMBINATORIAL PROPERTIES OFBINOMIAL EDGE IDEALS

This thesis consists of three parts that investigate the homological and combinatorial structure of binomial edge ideals.

The first part of this thesis is Chapter 3 and is based on the author’s paper [44]. We associate to every graph a linear program for packings of vertex disjoint paths. We show that the optimal primal and dual values of the corresponding integer program are the binomial grade and height of the binomial edge ideal of the graph. We deduce from this a new combinatorial characterization of graphs of König type and use it to show that all trees are of König type. The log canonical threshold and the F-threshold are important invariants associated to the singularities of a variety in characteristic 0 and characteristic p. We show that the optimal value of the linear program (computed over the rationals) agrees with both the F-threshold and the log canonical threshold of the binomial edge ideal if the graph is a block graph or of König type. We conjecture that this linear program computes the log canonical threshold of the binomial edge ideal of any graph. Our results resemble theorems on monomial ideals arising from hypergraphs due to Howald and others.

The second part of this thesis is Chapter 4 and is based on [42]. Since the introduction of binomial edge ideals JG by Herzog et al. and independently Ohtani, there has been significant interest in relating algebraic invariants of the binomial edge ideal with combinatorial invariants of the underlying graph G. Here, we take up a question considered by Herzog and Rinaldo regarding Castelnuovo–Mumford regularity of block graphs. To this end, we introduce a new invariant ν(G) associated to any simple graph G, defined as the maximal total length of a certain collection of induced paths within G subject to conditions on the induced subgraph. We prove that for any graph G, ν(G) ≤ reg(JG ) − 1, and that the length of a longest induced path of G is less than or equal to ν(G); this refines an inequality of Matsuda and Murai. We then investigate the question: when is ν(G) = reg(JG ) − 1? We prove that equality holds when G is closed; this gives a new characterization of a result of Ene and Zarojanu, and when G is bipartite and JG is Cohen-Macaulay; this gives a new characterization of a result of Jayanathan and Kumar. For a block graph G, we prove that ν(G) admits a combinatorial characterization independent of any auxiliary choices, and we prove that ν(G) = reg(JG ) − 1. This gives reg(JG ) a combinatorial interpretation for block graphs, and thus answers the question of Herzog and Rinaldo.

The third part of this thesis is Chapter 5 and is based on [43]. A famous theorem of Kalai and Meshulam is that reg(I + J) ≤ reg(I) + reg(J) − 1 for any squarefree monomial ideals I and J. This result was subsequently extended by Herzog to the case where I and J are any monomial ideals. In this chapter we conjecture that the Castelnuovo–Mumford regularity is subadditive on binomial edge ideals. Specifically, we propose that reg(JG ) ≤ reg(JH1 )+reg(JH2 )−1 whenever G, H1 , and H2 are graphs satisfying E(G) = E(H1 )∪E(H2 ) and J∗ is the associated binomial edge ideal. We prove a special case of this conjecture which strengthens the celebrated theorem of Malayeri–Madani–Kiani that reg(JG ) is bounded above by the minimal number of maximal cliques covering the edges of the graph G. From this special case we obtain a new upper bound for reg(JG ), namely that reg(JG ) ≤ ht(JG )+1. Our upper bound gives an analogue of the well-known result that reg(I(G)) ≤ ht(I(G))+1 where I(G) is the edge ideal of the graph G. We additionally prove that this conjecture holds for graphs admitting a combinatorial description for its Castelnuovo–Mumford regularity, that is for closed graphs, bipartite graphs with JG Cohen–Macaulay, and block graphs. Finally, we give examples to show that our new upper bound is incomparable with Malayeri–Madani–Kiani’s upper bound for reg(JG ) given by the size of a maximal clique disjoint set of edges.