science

Harris graph

Harris graph is a science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Harris graph rather than just read about it. In short: In graph theory, a Harris graph is defined as an Eulerian, tough, non-Hamiltonian graph. Harris graphs were introduced in 2013 when, at the University of Michigan, Harris Spungen conjectured that any graph which is both tough and Eulerian is sufficiently Hamiltonian.

Key takeaways

Harris graph belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Harris graph to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Harris graph from memory before moving on to harder problems.

Reference excerpt

In graph theory, a Harris graph is defined as an Eulerian, tough, non-Hamiltonian graph. Harris graphs were introduced in 2013 when, at the University of Michigan, Harris Spungen conjectured that any graph which is both tough and Eulerian is sufficiently Hamiltonian. However, Douglas Shaw disproved this conjecture, discovering a counterexample with order 9 and size 14. Currently, there are 241,375 known Harris graphs. The minimal Harris graph, the Hirotaka graph, has order 7 and size 12. Harris graphs can be constructed by adding barnacles or grafting smaller Harris graphs, enabling larger graphs while preserving their properties. Notable types include the minimal Hirotaka graph, the barnacle-free Lopez graph, and the Shaw graph, each showcasing unique structural features in graph theory. Harris graphs are valuable for teaching graph theory due to their accessible methods for finding and verifying them. They offer a balanced challenge, fostering creativity, teamwork, and problem-solving as students collaborate to explore solutions.

History After Harris Spungen made his conjecture in 2013, Doug Shaw shortly discovered a counterexample, the Harris graph. Jayna Fishman and Elizabeth Petrie found two more Harris graphs in the same year. Over the next few years, three more Harris graphs were discovered, until Hirotaka Yoneda discovered what was thought to be the minimal Harris graph in 2018. In 2023, and the Hirotaka graph was proven to be unique by code written by Shubhra Mishra and Marco Troper. The number of Harris graphs with n vertices was also made into an OEIS sequence.

Construction

Flowering a Harris graph A k-barnacle is a path of length k between two nodes where every node on the path has degree 2. Flowering is the process of adding a 2-barnacle between two nodes on the shortest path between two odd-degree nodes. Flowering a tough, non-Hamiltonian graph that has an even number of nodes with odd degrees produces a Harris graph. Adding a 2-barnacle to a graph preserves its toughness while making it more difficult to be Hamiltonian. Furthermore, because a graph cannot have an odd number of vertices with odd degrees, the process of flowering can transform any non-Hamiltonian, tough graph into an Eulerian one as well.

Grafting two Harris graphs into one A 5-wheel is added between one edge in one Harris graph and another edge in another Harris graph, and two nodes from each 5-wheel are connected to each other that were not connected to the original graph. Since adding the connections and the 5-wheel does not cause the graph to be Hamiltonian, non-Eulerian, or not tough if it already met those conditions, the result will be one Harris graph.

Replacing edges with barnacles Replacing an edge from an existing Harris graph with a 2-barnacle creates a Harris graph since all old degrees will be preserved, while the barnacle has a degree of 2 by definition, so the graph is still Eulerian. Since it is now even harder for the graph to be Hamiltonian, and since the graph's toughness remains the same, adding a barnacle anywhere keeps the graph Eulerian, tough, and non-Hamiltonian.

Types

The Hirotaka graph, with 7 and size 12, is the Harris graph with the smallest order. The first appearance of this graph was as an example of a non-Hamiltonian, tough graph. Douglas Shaw proved it to be minimal by showing all Eulerian graphs of order 6 or lower were not Hamiltonian and tough. Java code written by Shubhra Mishra and Marco Troper proved it unique. The first Harris graph discovered was the Shaw graph, which has order 9 and size 14. This graph served as the counterexample to Harris Spungen's 2013 conjecture. The minimal barnacle-free Harris graph, or the Lopez graph, has order 13 and size 33. It was constructed to address a conjecture that barnacle-free Harris graphs do not exist.

Applications Harris graphs are particularly valuable in teaching graph theory because they possess easily understandable properties and methods for finding and verifying them. They offer an ideal balance between challenge and accessibility, making them an engaging problem for students at various levels. Working with Harris graphs encourages students to experiment with different concepts and solutions, promoting creativity and mathematical thinking. This process keeps students engaged and collaborating with each other, as they often work together to verify potential solutions, enhancing teamwork and collective problem-solving skills.

References

Illustrations

Worked examples

Example 1 — a first encounter with Harris graph

Start with the simplest possible case. Write down what Harris graph claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Harris graph before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Harris graph ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Harris graph

In research: Harris graph appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Harris graph in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Harris graph is common in secondary-school and first-year university syllabi. It links to neighbouring topics Graph families, so understanding it makes those chapters shorter.
In everyday life: Look for Harris graph outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Harris graph in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Harris graph means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Harris graph out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Harris graph in simple terms?

Why does Harris graph matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Harris graph?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Harris graph.