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...a cycle of length near <math>n^e</math> but some graphs do not have longer cycles. More precisely, for any ordering of the graphs in <math>{\mathcal F}</math ...ies of graphs that always contain a Hamiltonian or near-Hamiltonian cycle, and 0 for families of graphs in which the longest cycle length can be smaller t ...

{{Short description|Planar bipartite graph with 25 vertices and 31 edges}} ...herGraph|title=Walther Graph}}</ref> It has [[chromatic index]] 3, girth 3 and diameter 8. ...

{{Short description|On Hamiltonian cycles in planar graphs}} ...rinberg 5CEC Nonhamiltonian graph.svg|thumb|A graph that can be proven non-Hamiltonian using Grinberg's theorem]] ...

[[File:Walecki decomposition.svg|thumb|upright=1.35|Walecki's Hamiltonian decomposition of the complete graph <math>K_9</math>]] ...amiltonian decompositions have been studied both for [[undirected graph]]s and for [[directed graph]]s. ...

In mathematics, the '''Hamiltonian cycle polynomial''' of an ''n''×''n''-matrix is a polynomial in its entries ...option useful, in a number of cases, for determining the existence of a [[Hamiltonian cycle]] in a [[directed graph]]. ...

...raph}}</ref> It has [[chromatic number]] 3, [[chromatic index]] 3, girth 4 and diameter 8. ...nterexample to [[Tait's conjecture]] that every 3-regular polyhedron has a Hamiltonian cycle.<ref>{{citation|first=P. G.|last=Tait|authorlink=P. G. Tait|title=Lis ...

{{Short description|Graph containing cycles of all possible lengths}} [[File:Pancyclic octahedron.svg|thumb|Cycles of all possible lengths in the graph of an [[octahedron]], showing it to be ...

{{Short description|Bipartite non-Hamiltonian polyhedral graph}} *[[Hamiltonian cycle|non-Hamiltonian]] ...

..., Y.N.; Grover, W.D.; "p-Cycles: An overview," IEEE Communications Surveys and Tutorials, vol.12, no.1, pp.97-111, First Quarter 2010</ref><ref name="Anno ...n a case of a link failure to switch traffic to a pre-planned cycle (path) and recover, as it is demonstrated in Figure 2. ...

{{Short description|Theorem on Hamiltonian graphs}} ...er's theorem.svg|thumb|360px|A 2-vertex-connected graph, its square, and a Hamiltonian cycle in the square]] ...

{{short description|Graph made from vertices and edges of a convex polyhedron}} ...is the [[undirected graph]] formed from the [[Vertex (geometry)|vertices]] and [[Edge (geometry)|edges]] of a [[convex polyhedron]]. Alternatively, in pur ...

..., and to several related problems including the [[Hamiltonian path problem|Hamiltonian cycle problem]], in [[exponential time]]. ==Algorithm description and motivation== ...

...hs without negative-weight cycles, the longest path problem is [[NP-hard]] and the decision version of the problem, which asks whether a path exists of at ...timization: Polyhedra and Efficiency, Volume 1|volume=24|series=Algorithms and Combinatorics|first=Alexander|last=Schrijver|authorlink=Alexander Schrijver ...

[[File:Polytrees in a tournament.svg|thumb|upright=1.3|A 6-vertex tournament, and copies of every 4-vertex oriented tree within it.]] ...lished citations given by Kühn et al. are to {{harvtxt|Reid|Wormald|1983}} and {{harvtxt|Wormald|1983}}. {{harvtxt|Wormald|1983}} cites the conjecture as ...

...directed graph]] constructed by assigning a [[real number]] to each vertex and connecting two vertices by an edge when their numbers are within one unit o ...for the indifference graphs: the claw, sun, and net (top, left–right) and cycles of length four or more (bottom)]] ...

...erfere with each other at any of their common neighbors,<ref name="ah00"/> and to find [[graph drawing]]s with high [[angular resolution (graph drawing)|a Both the [[chromatic number]] and the [[degeneracy (graph theory)|degeneracy]] of the {{mvar|k}}th power of a ...

...econd edition, prepared with the assistance of Volker Kaibel, Victor Klee, and [[Günter M. Ziegler]], was published by [[Springer-Verlag]] in 2003, as vol ...topes with a number of vertices only slightly higher than their dimension, and [[neighborly polytope]]s.{{r|mcmullen|zvonkin}} ...

...[[Set (mathematics)|set]] (non-empty) ''V''(''G''), edge set ''E''(''G'') and [[Degree (graph theory)|maximum degree]] Δ(''G''). ...that vertex ''v'' is incident to edge ''e''. Two incidences (''v'', ''e'') and (''u'', ''f'') are said to be '''adjacent''' or '''neighboring''' if one of ...

*No two adjacent vertices have the same color, and ...dre|last4 = Szemerédi|citeseerx = 10.1.1.224.5588| s2cid=18721867 }}</ref> and for finding optimal colorings of special classes of graphs, but the more ge ...

...t1=Anders |last2=Mølmer |first2=Klaus |date=2000-07-18 |title=Entanglement and quantum computation with ions in thermal motion |url=https://link.aps.org/d ...s are prepared by illuminating ions with a bichromatic light field. Mølmer and Sørensen identified two regimes in which this is possible: ...