7,720 research outputs found
Graphs with few Hamiltonian Cycles
We describe an algorithm for the exhaustive generation of non-isomorphic
graphs with a given number of hamiltonian cycles, which is especially
efficient for small . Our main findings, combining applications of this
algorithm and existing algorithms with new theoretical results, revolve around
graphs containing exactly one hamiltonian cycle (1H) or exactly three
hamiltonian cycles (3H). Motivated by a classic result of Smith and recent work
of Royle, we show that there exist nearly cubic 1H graphs of order iff is even. This gives the strongest form of a theorem of Entringer and
Swart, and sheds light on a question of Fleischner originally settled by
Seamone. We prove equivalent formulations of the conjecture of Bondy and
Jackson that every planar 1H graph contains two vertices of degree 2, verify it
up to order 16, and show that its toric analogue does not hold. We treat
Thomassen's conjecture that every hamiltonian graph of minimum degree at least
contains an edge such that both its removal and its contraction yield
hamiltonian graphs. We also verify up to order 21 the conjecture of Sheehan
that there is no 4-regular 1H graph. Extending work of Schwenk, we describe all
orders for which cubic 3H triangle-free graphs exist. We verify up to order
Cantoni's conjecture that every planar cubic 3H graph contains a triangle,
and show that there exist infinitely many planar cyclically 4-edge-connected
cubic graphs with exactly four hamiltonian cycles, thereby answering a question
of Chia and Thomassen. Finally, complementing work of Sheehan on 1H graphs of
maximum size, we determine the maximum size of graphs containing exactly one
hamiltonian path and give, for every order , the exact number of such graphs
on vertices and of maximum size.Comment: 29 pages; to appear in Mathematics of Computatio
Hamilton cycles in graphs and hypergraphs: an extremal perspective
As one of the most fundamental and well-known NP-complete problems, the
Hamilton cycle problem has been the subject of intensive research. Recent
developments in the area have highlighted the crucial role played by the
notions of expansion and quasi-randomness. These concepts and other recent
techniques have led to the solution of several long-standing problems in the
area. New aspects have also emerged, such as resilience, robustness and the
study of Hamilton cycles in hypergraphs. We survey these developments and
highlight open problems, with an emphasis on extremal and probabilistic
approaches.Comment: to appear in the Proceedings of the ICM 2014; due to given page
limits, this final version is slightly shorter than the previous arxiv
versio
On some intriguing problems in Hamiltonian graph theory -- A survey
We survey results and open problems in Hamiltonian graph theory centred around three themes: regular graphs, -tough graphs, and claw-free graphs
Proof of Koml\'os's conjecture on Hamiltonian subsets
Koml\'os conjectured in 1981 that among all graphs with minimum degree at
least , the complete graph minimises the number of Hamiltonian
subsets, where a subset of vertices is Hamiltonian if it contains a spanning
cycle. We prove this conjecture when is sufficiently large. In fact we
prove a stronger result: for large , any graph with average degree at
least contains almost twice as many Hamiltonian subsets as ,
unless is isomorphic to or a certain other graph which we
specify.Comment: 33 pages, to appear in Proceedings of the London Mathematical Societ
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