graph theory In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph ...

, a Harris

graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discret ...

is defined as an Eulerian, tough, non-

Hamiltonian graph In the mathematical field of graph theory, a Hamiltonian path (or traceable path) is a path in an undirected or directed graph that visits each vertex exactly once. A Hamiltonian cycle (or Hamiltonian circuit) is a cycle that visits each verte ...

. Harris graphs were introduced in 2013 when, at the

University of Michigan The University of Michigan (U-M, U of M, or Michigan) is a public university, public research university in Ann Arbor, Michigan, United States. Founded in 1817, it is the oldest institution of higher education in the state. The University of Mi ...

, Harris Spungen

conjecture In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), ha ...

d that any graph which is both tough and Eulerian is sufficiently Hamiltonian. However, Douglas Shaw disproved this

, discovering a

counterexample A counterexample is any exception to a generalization. In logic a counterexample disproves the generalization, and does so rigorously in the fields of mathematics and philosophy. For example, the fact that "student John Smith is not lazy" is a c ...

with

order Order, ORDER or Orders may refer to: * A socio-political or established or existing order, e.g. World order, Ancien Regime, Pax Britannica * Categorization, the process in which ideas and objects are recognized, differentiated, and understood ...

9 and

size Size in general is the Magnitude (mathematics), magnitude or dimensions of a thing. More specifically, ''geometrical size'' (or ''spatial size'') can refer to three geometrical measures: length, area, or volume. Length can be generalized ...

14. Currently, there are 241,375 known Harris graphs. The minimal Harris graph, the Hirotaka graph, has order 7 and size 12. Harris

graphs Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties * Graph (topology), a topological space resembling a graph in the sense of discre ...

can be constructed by adding barnacles or grafting smaller Harris

, enabling larger graphs while preserving their properties. Notable types include the minimal Hirotaka

, the barnacle-free Lopez

, and the Shaw

, each showcasing unique structural features in

. Harris graphs are valuable for teaching

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

in 2013, Doug Shaw shortly discovered a

, 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 The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while researching at AT&T Labs. He transferred the intellectual property and hosting of the OEIS to th ...

sequence.

Construction

Flowering a Harris graph

A k-barnacle is a

path A path is a route for physical travel – see Trail. Path or PATH may also refer to: Physical paths of different types * Bicycle path * Bridle path, used by people on horseback * Course (navigation), the intended path of a vehicle * Desir ...

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-

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 A wheel is a rotating component (typically circular in shape) that is intended to turn on an axle Bearing (mechanical), bearing. The wheel is one of the key components of the wheel and axle which is one of the Simple machine, six simple machin ...

is added between one

edge Edge or EDGE may refer to: Technology Computing * Edge computing, a network load-balancing system * Edge device, an entry point to a computer network * Adobe Edge, a graphical development application * Microsoft Edge, a web browser developed by ...

in one Harris graph and another edge in another Harris graph, and two nodes from each 5-wheel are

connected Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular'' * '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film * ''Connected'' (2015 TV ...

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 In materials science and metallurgy, toughness is the ability of a material to absorb energy and plastically deform without fracturing.

Types

The Hirotaka graph, with 7 and

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 Java is one of the Greater Sunda Islands in Indonesia. It is bordered by the Indian Ocean to the south and the Java Sea (a part of Pacific Ocean) to the north. With a population of 156.9 million people (including Madura) in mid 2024, proje ...

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

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

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