Hilbert Bases 

Examples of Non‐Hilbert Matroids 

Marko Mitrovic 

Acknowledgements •  Supervisor: Dr. Luis Goddyn •  Dr. Gordon Royle (University of Western

Australia) •  Dr. Stefan van Zwam (Princeton

University) •  Special thanks to Dr. Tony Huynh and

Tanmay Deshpande

Hilbert Bases and Graphs 

Extension to Non‐Hilbert Matroids 

A matroid can be viewed as a generalization of a graph, where the analogue of a cut-set is called a cocircuit. Below are the steps we took to find all non-Hilbert matroids of at most 8 elements:

1.  Generate a list of all matroids up to and including 8 elements.

2.  Generate the list of cocircuits for each matroid. 3.  For each matroid, compile the characteristic

vectors of the cocircuits into a .in file that Normaliz can read.

4.  Run each .in file through Normaliz, resulting in a .out file for each .in file.

5.  At this point, every matroid will have an associated .out file. Read through each .out file to determine whether or not the associated matroid is Hilbert.

A cut of a graph is a partition of the vertices into 2 disjoint subsets. Two possible cuts of G are shown below.

Figure 1 Figure 2

Examples of Non‐Hilbert Graphs 

We define H to be the set of all graphs G such that the set of characteristic vectors of cuts of G is a Hilbert basis.

If we input the characteristic vectors of all the cuts of a particular graph G into the Normaliz software package, it will determine whether or not G ∈H .