** Harris Graphs: Tough Eulerian Graphs that are not Hamiltonian**

**Harris Graphs: Tough Eulerian Graphs that are not Hamiltonian**

Eulerian graph: a connected graph where all vertices have an even degree.

Tough graph: A graph G is tough if, for every subset S of the vertices of G, the number of components of G \ S is at most the number of vertices in S (more info here).

Hamiltonian cycle: a path that ends at the vertex it started and visits all vertices exactly once (The first vertex is technically visited twice - once in the beginning and once at the end).

**Feel free to play around with the graphs below to test the G \ S condition. To delete a vertex, right click on it and hit delete. If you find a new Harris Graph or believe that one of the graphs listed here isn't a Harris graph, fill out the Prospective Harris Graph or Error Reporting Form respectively.**

**Doug Shaw-9-14**

**Doug Shaw-9-14**

Discovered in 2013.

**Jayna Fishman-Elizabeth Petrie-9-15**

**Jayna Fishman-Elizabeth Petrie-9-15**

Discovered in 2013.

**Jayna Fishman-Elizabeth Petrie****-12-20**

**Jayna Fishman-Elizabeth Petrie**

**-12-20**

Discovered in 2013.

**Jungmin Kwon-Jingheng (Sunny) Li-Minseo Son-Ruihan Wang-15-27**

**Jungmin Kwon-Jingheng (Sunny) Li-Minseo Son-Ruihan Wang-15-27**

Discovered in 2016.

**Alex Leaf-9-15**

**Alex Leaf-9-15**

Discovered in 2016. Simplified version of ZackJoseph-12-18.

**Doug Shaw-18-30**

**Doug Shaw-18-30**

Discovered in 2016.

**Hirotaka-7-12**

**Hirotaka-7-12**

Discovered in 2018. Minimal Harris graph.

**2019Students-8-14**

**2019Students-8-14**

Discovered in 2019.

**Zach DeVivo-Marco Troper-10-18**

**Zach DeVivo-Marco Troper-10-18**

Discovered in 2022.

**Shubhra Mishra****-****11****-****20**

**Shubhra Mishra**

**-**

**11**

**-**

**20**

Discovered in 2022.

**Zhennong Li****-1****3****-2****3**

**Zhennong Li**

**-1**

**3**

**-2**

**3**

Discovered in 2022.

**Jacob-15-25**

**Jacob-15-25**

Discovered in 2022.