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The Hamiltonian cycle in the Cayley graph of the symmetric group generated by the Steinhaus–Johnson–Trotter algorithm Wheel diagram of all permutations of length = generated by the Steinhaus-Johnson-Trotter algorithm, where each permutation is color-coded (1=blue, 2=green, 3=yellow, 4=red).
Barnette's conjecture is an unsolved problem in graph theory, a branch of mathematics, concerning Hamiltonian cycles in graphs. It is named after David W. Barnette, a professor emeritus at the University of California, Davis; it states that every bipartite polyhedral graph with three edges per vertex has a Hamiltonian cycle.
It implies that computing, up to the -th power of , the Hamiltonian cycle polynomial of a unitary n×n-matrix over the infinite extension of any ring of characteristic q (not necessarily prime) by the formal variable is a # P-complete problem if isn't 2 and computing the Hamiltonian cycle polynomial of a -semi-unitary matrix (i.e. an n×n ...
A Hamiltonian cycle, Hamiltonian circuit, vertex tour or graph cycle is a cycle that visits each vertex exactly once. A graph that contains a Hamiltonian cycle is called a Hamiltonian graph . Similar notions may be defined for directed graphs , where each edge (arc) of a path or cycle can only be traced in a single direction (i.e., the vertices ...
A graph that can be proven non-Hamiltonian using Grinberg's theorem. In graph theory, Grinberg's theorem is a necessary condition for a planar graph to contain a Hamiltonian cycle, based on the lengths of its face cycles. If a graph does not meet this condition, it is not Hamiltonian.
A graph G is subhamiltonian if G is a subgraph of another graph aug(G) on the same vertex set, such that aug(G) is planar and contains a Hamiltonian cycle.For this to be true, G itself must be planar, and additionally it must be possible to add edges to G, preserving planarity, in order to create a cycle in the augmented graph that passes through each vertex exactly once.
The line graphs of cubic graphs are 4-regular and have a Hamiltonian decomposition if and only if the underlying cubic graph has a Hamiltonian cycle. [12] [13] As a consequence, Hamiltonian decomposition remains NP-complete for classes of graphs that include line graphs of hard instances of the Hamiltonian cycle problem. For instance ...
Hamiltonian cycle in M 4 (Grötzsch graph) If G has chromatic number k, then μ(G) has chromatic number k + 1 (Mycielski 1955). If G is triangle-free, then so is μ(G) (Mycielski 1955). More generally, if G has clique number ω(G), then μ(G) has clique number of the maximum among 2 and ω(G). (Mycielski 1955)