Proves if-and-only-if equivalences for toric ring normality and quadratic toric ideal generation between anti-blocking lattice polytopes and their unconditional reflections, plus a graph-theoretic characterization of quadratic symmetric stable set ideals.
Kempe equivalence and quadratic toric rings
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abstract
Kempe equivalence is a classical and fundamental notion in graph coloring theory. In the present paper we establish a connection between Kempe equivalence and quadratic stable set ring, which are toric rings associated to graphs. In fact, we characterize when the stable set ring of a graph is quadratic by using Kempe equivalence. As an application, we relate our theorem to the theory of perfectly contractile graphs, a hereditary subclass of perfect graphs introduced by Bertschi. In particular, our characterization implies that the conjecture of Everett and Reed on perfectly contractile graphs entails the conjecture of the authors and Shibata on quadratic stable set rings. Furthermore, we show that the stable set rings of several important subclasses of perfectly contractile graphs including weakly chordal graphs are quadratic. Finally, we propose a new combinatorial conjecture characterizing perfectly contractile graphs purely in terms of Kempe equivalence on replication graphs.
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Gorenstein simplices with the given h*-polynomial are classified up to unimodular equivalence by strict divisor chains in the divisor lattice of v, yielding an explicit counting formula.
Toric ideals of matching polytopes for bipartite graphs have generators of degree at most 3, shown equivalent to an edge-coloring result, with a new characterization for the quadratic case.
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Algebraic aspects of unconditional lattice polytopes
Proves if-and-only-if equivalences for toric ring normality and quadratic toric ideal generation between anti-blocking lattice polytopes and their unconditional reflections, plus a graph-theoretic characterization of quadratic symmetric stable set ideals.
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Classification and counting of Gorenstein simplices with $h^*$-polynomial $1+t^k+\cdots+t^{(v-1)k}$
Gorenstein simplices with the given h*-polynomial are classified up to unimodular equivalence by strict divisor chains in the divisor lattice of v, yielding an explicit counting formula.
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Toric ideal of matching polytopes and edge colorings
Toric ideals of matching polytopes for bipartite graphs have generators of degree at most 3, shown equivalent to an edge-coloring result, with a new characterization for the quadratic case.