Fleischner's theorem



In graph theory, a branch of mathematics, Fleischner's theorem gives a sufficient condition for a graph to contain a Hamiltonian cycle. It states that, if $$G$$ is a 2-vertex-connected graph, then the square of $$G$$ is Hamiltonian. It is named after Herbert Fleischner, who published its proof in 1974.

Definitions and statement
An undirected graph $$G$$ is Hamiltonian if it contains a cycle that touches each of its vertices exactly once. It is 2-vertex-connected if it does not have an articulation vertex, a vertex whose deletion would leave the remaining graph disconnected. Not every 2-vertex-connected graph is Hamiltonian; counterexamples include the Petersen graph and the complete bipartite graph $$K_{2,3}$$.

The square of $$G$$ is a graph $$G^2$$ that has the same vertex set as $$G$$, and in which two vertices are adjacent if and only if they have distance at most two in $$G$$. Fleischner's theorem states that the square of a finite 2-vertex-connected graph with at least three vertices must always be Hamiltonian. Equivalently, the vertices of every 2-vertex-connected graph $$G$$ may be arranged into a cyclic order such that adjacent vertices in this order are at distance at most two from each other in $$G$$.

Extensions
In Fleischner's theorem, it is possible to constrain the Hamiltonian cycle in $$G^2$$ so that for given vertices $$v$$ and $$w$$ of $$G$$ it includes two edges of $$G$$ incident with $$v$$ and one edge of $$G$$ incident with $w$. Moreover, if $$v$$ and $$w$$ are adjacent in $$G$$, then these are three different edges of $$G$$.

In addition to having a Hamiltonian cycle, the square of a 2-vertex-connected graph $$G$$ must also be Hamiltonian connected (meaning that it has a Hamiltonian path starting and ending at any two designated vertices) and 1-Hamiltonian (meaning that if any vertex is deleted, the remaining graph still has a Hamiltonian cycle). It must also be vertex pancyclic, meaning that for every vertex $$v$$ and every integer $$k$$ with $$3\le k\le|V(G)$$, there exists a cycle of length $$k$$ containing $$v$$.

If a graph $$G$$ is not 2-vertex-connected, then its square may or may not have a Hamiltonian cycle, and determining whether it does have one is NP-complete.

An infinite graph cannot have a Hamiltonian cycle, because every cycle is finite, but Carsten Thomassen proved that if $$G$$ is an infinite locally finite 2-vertex-connected graph with a single end then $$G^2$$ necessarily has a doubly infinite Hamiltonian path. More generally, if $$G$$ is locally finite, 2-vertex-connected, and has any number of ends, then $$G^2$$ has a Hamiltonian circle. In a compact topological space formed by viewing the graph as a simplicial complex and adding an extra point at infinity to each of its ends, a Hamiltonian circle is defined to be a subspace that is homeomorphic to a Euclidean circle and covers every vertex.

Algorithms
The Hamiltonian cycle in the square of an $$n$$-vertex 2-connected graph can be found in linear time, improving over the first algorithmic solution by Lau of running time $$O(n^2)$$. Fleischner's theorem can be used to provide a 2-approximation to the bottleneck traveling salesman problem in metric spaces.

History
A proof of Fleischner's theorem was announced by Herbert Fleischner in 1971 and published by him in 1974, solving a 1966 conjecture of Crispin Nash-Williams also made independently by L. W. Beineke and Michael D. Plummer. In his review of Fleischner's paper, Nash-Williams wrote that it had solved "a well known problem which has for several years defeated the ingenuity of other graph-theorists".

Fleischner's original proof was complicated. Václav Chvátal, in the work in which he invented graph toughness, observed that the square of a $$k$$-vertex-connected graph is necessarily $$k$$-tough; he conjectured that 2-tough graphs are Hamiltonian, from which another proof of Fleischner's theorem would have followed. Counterexamples to this conjecture were later discovered, but the possibility that a finite bound on toughness might imply Hamiltonicity remains an important open problem in graph theory. A simpler proof both of Fleischner's theorem, and of its extensions by, was given by , and another simplified proof of the theorem was given by.