Abstract
The Ehrhart ring of the edge polytope \({\mathcal{P}_G}\) for a connected simple graph G is known to coincide with the edge ring of the same graph if G satisfies the odd cycle condition. This paper gives for a graph which does not satisfy the condition, a generating set of the defining ideal of the Ehrhart ring of the edge polytope, described by combinatorial information of the graph. From this result, two factoring properties of the Ehrhart series are obtained; the first one factors out bipartite biconnected components, and the second one factors out a even cycle which shares only one edge with other part of the graph. As an application of the factoring properties, the root distribution of Ehrhart polynomials for bipartite polygon trees is determined.
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Acknowledgments
I would like to thank Hidefumi Ohsugi for kind and helpful comments, on an early stage in particular. Takayuki Hibi gave me several invaluable and encouraging comments on the first draft. Discussions with Akihiro Higashitani greatly motivated me to think factorization properties. Suggestions from Takeaki Uno about choice of graphs were very useful for Sect. 4.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Matsui, T. Ehrhart Series for Connected Simple Graphs. Graphs and Combinatorics 29, 617–635 (2013). https://doi.org/10.1007/s00373-011-1126-y
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DOI: https://doi.org/10.1007/s00373-011-1126-y