Computational Breakthroughs in Graph Theory: Simon MacLean’s Work on Tarsi and Stanley Conjectures
Simon MacLean, a senior Mathematics major and Computer Science minor at Santa Clara University, is using high-performance computing to tackle longstanding problems in theoretical mathematics. Working with Professor Shamil Asgarli, Simon leverages the WAVE HPC cluster to explore complex questions in graph theory, the study of mathematical structures made of nodes and edges.
Beyond "Pen and Paper"
Simon focuses on two challenging problems: Tarsi's Conjecture and Stanley's Tree Isomorphism Conjecture. Tarsi's Conjecture, a type of graph coloring problem, asks whether a special class of graphs can always be colored with five colors so that no two connected nodes share the same color. Using WAVE, Simon was able to scan large datasets of graphs, surfacing a specific graph that had evaded his theoretical filters.
The graph Simon identified using WAVE, showing the edge 5-cut that challenged existing theoretical restrictions.
This graph contained an edge 5-cut, a configuration where five edges split the graph into two parts. Identifying this structure helped Simon sharpen his understanding of which graphs satisfy the conditions of the conjecture.
"Using WAVE, using a ton of CPUs at once, I was able to find one graph that had slipped by the filters I'd had so far," Simon explains.
Breaking the Memory Barrier
Stanley's Tree Isomorphism Conjecture proposes that no two distinct trees—graphs without cycles—share the same identifying data. Computational verification of this conjecture had previously reached n = 29 nodes, after which researchers hit memory limits at n = 30, because the number of distinct trees grows into the billions.
To tackle this, Simon leveraged a specialized WAVE node with two terabytes of RAM, providing the memory capacity needed for large-scale computation.
"There's this node on the WAVE that has two terabytes of memory, which is just an insane amount of memory really," Simon says.
Algorithmic Innovation
Rather than being limited by memory constraints himself, Simon introduced a new computational approach. By processing tree data in independent partitions instead of loading the entire dataset into memory, he extended the verification of Stanley's conjecture beyond the previous computational barrier.
This method allowed him to successfully verify the conjecture for n = 30 through 33 nodes, and he is currently working on n = 34.
"Even if I can just improve it by a few nodes and post a note somewhere, that's the kind of thing that would be referenced in a bunch of papers in the future," he notes.
Impact and Future Goals
Simon’s computational work provides a new baseline for graph theory research, as subsequent papers often begin by citing the highest verified node count. His achievements also contributed to his acceptance into several PhD programs, and he is currently considering Duke University, where he may pursue his doctorate in mathematics this fall.
Recognition
Simon’s research at Santa Clara University has been recognized with several awards:
- George W. Evans II Memorial Prize for top performance in a major math competition
- Marjorie Louise Woodard Evans & George W. Evans II Memorial Prizes for Mathematical Writing and Research
- Rick Scott Memorial Scholarship
- De Novo Fellowship Award
- REAL Program Stipend
Through a combination of mathematical insight, high-performance computing, and creative problem-solving, Simon is demonstrating how modern technology can push the boundaries of theory itself.
HPC at a Glance
|
Field |
Graph theory (pure mathematics) |
|
Focus |
Tarsi’s Conjecture; Stanley’s Tree Isomorphism Conjecture |
|
HPC Used |
WAVE cluster, high-memory node (2 TB RAM), parallel CPUs |
|
Researcher |
Simon MacLean, undergraduate researcher |
|
Key Result |
Verified Stanley’s conjecture to n = 33 (prev. n = 29) |
Based on an interview conducted by Ella Griffin, WAVE Student Marketing Assistant, on February 13, 2026.