Hamiltonian paths and cycles

A **Hamiltonian path** is a specific type of path in a graph that visits each vertex exactly once. In other words, it is a trail in which every node (or vertex) of the graph is included exactly one time. A **Hamiltonian cycle** (or Hamiltonian circuit) is a special case of a Hamiltonian path where the path starts and ends at the same vertex, thus forming a closed loop that visits every vertex once.

Barnette's conjecture is a proposition in the field of combinatorial geometry, specifically concerning polyhedra. It states that for a polyhedron with \( n \) vertices, the number of faces \( f \) must satisfy the inequality: \[ f \leq 2n - 4 \] This conjecture essentially posits an upper bound on the number of faces in a convex polyhedron based on its number of vertices.

Hamiltonian completion is a concept in graph theory related to the idea of completing a given graph into a Hamiltonian graph. A Hamiltonian graph is one that contains a Hamiltonian cycle, which is a cycle that visits every vertex in the graph exactly once and returns to the starting vertex. Hamiltonian completion specifically deals with taking an incomplete graph (one that may not be Hamiltonian) and determining whether it is possible to add a certain number of edges to make it Hamiltonian.

The Herschel graph, also known as the Herschel-Dickson graph, is a specific type of undirected graph that is notable in the study of mathematical graphs and combinatorial design. It is a bipartite graph that is defined as follows: 1. **Vertices**: The Herschel graph consists of 14 vertices. It can be visualized as having two sets of vertices: - One set consists of 7 vertices (usually denoted as \( U \)).

A hypohamiltonian graph is a type of graph in graph theory that is defined as follows: a graph \( G \) is considered hypohamiltonian if it is not Hamiltonian (i.e., it does not contain a Hamiltonian circuit) but the removal of any single vertex from \( G \) results in a graph that is Hamiltonian.

A **pancyclic graph** is a type of graph in graph theory that contains cycles of all possible lengths from 3 up to the maximum length that is less than or equal to the number of vertices in the graph.

The term "shortness exponent" isn't widely known or defined within established scientific literature as of my last update. However, it's possible that it may refer to a concept in a specialized area of research, possibly in fields like physics, mathematics, or data analysis, where exponents are used to characterize statistical properties of distributions or phenomena. If you're referring to a concept in a specific context (e.g.

A **subhamiltonian graph** is a type of graph in the field of graph theory. Specifically, a subhamiltonian graph is one that contains a Hamiltonian path but not necessarily a Hamiltonian cycle. In other words, it is possible to traverse all vertices in the graph exactly once (the definition of a Hamiltonian path), but it may not be possible to return to the starting vertex without repeating any vertices (which would be needed for a Hamiltonian cycle).

Tait's conjecture is a statement in graph theory and topology related to the study of knot diagrams. Proposed by the Scottish mathematician Peter Tait in the 19th century, the conjecture specifically pertains to the number of crossings in alternating knot diagrams.

A Walther graph is a type of graph that arises in the context of graph theory, particularly in the study of order types and combinatorial structures. It is constructed using the points of a finite projective plane. Specifically, a Walther graph is formed from a set of points and lines in a projective plane, where the vertices of the graph represent points, and edges connect pairs of vertices if the corresponding points lie on the same line.