Annals of Combinatorics

, Volume 17, Issue 1, pp 71–103 | Cite as

Ideals of Graph Homomorphisms

  • Alexander Engström
  • Patrik Norén


In combinatorial commutative algebra and algebraic statistics many toric ideals are constructed from graphs. Keeping the categorical structure of graphs in mind we give previous results a more functorial context and generalize them by introducing the ideals of graph homomorphisms. For this new class of ideals we investigate how the topology of the graphs influences the algebraic properties. We describe explicit Gröbner bases for several classes, generalizing results by Hibi, Sturmfels, and Sullivant. One of our main tools is the toric fiber product, and we employ results by Engström, Kahle, and Sullivant. The lattice polytopes defined by our ideals include important classes in optimization theory, as the stable set polytopes.

Mathematics Subject Classification

05C60 68W30 13P25 13P10 62H17 


graph homomorphisms toric ideals Gröbner bases algebraic statistics structural graph theory 


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Copyright information

© Springer Basel 2012

Authors and Affiliations

  1. 1.Department of MathematicsAalto UniversityAaltoFinland

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