25 research outputs found
Hamiltonicity in connected regular graphs
In 1980, Jackson proved that every 2-connected -regular graph with at most
vertices is Hamiltonian. This result has been extended in several papers.
In this note, we determine the minimum number of vertices in a connected
-regular graph that is not Hamiltonian, and we also solve the analogous
problem for Hamiltonian paths. Further, we characterize the smallest connected
-regular graphs without a Hamiltonian cycle.Comment: 5 page
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
Cycles containing all vertices of maximum degree
For a graph G and an integer k, denote by Vk the set {v ε V(G) | d(v) ≥ k}. Veldman proved that if G is a 2-connected graph of order n with n ≤ 3k - 2 and |Vk| ≤ k, then G has a cycle containing all vertices of Vk. It is shown that the upper bound k on |Vk| is close to best possible in general. For the special case k = δ(G), it is conjectured that the condition |Vk| ≤ k can be omitted. Using a variation of Woodall's Hopping Lemma, the conjecture is proved under the additional condition that n ≤ 2δ(G) + δ(G) + 1. This result is an almost-generalization of Jackson's Theorem that every 2-connected k-regular graph of order n with n ≤ 3k is hamiltonian. An alternative proof of an extension of Jackson's Theorem is also presented
Hamiltonian cycles in Cayley graphs of imprimitive complex reflection groups
Generalizing a result of Conway, Sloane, and Wilkes for real reflection
groups, we show the Cayley graph of an imprimitive complex reflection group
with respect to standard generating reflections has a Hamiltonian cycle. This
is consistent with the long-standing conjecture that for every finite group, G,
and every set of generators, S, of G the undirected Cayley graph of G with
respect to S has a Hamiltonian cycle.Comment: 15 pages, 4 figures; minor revisions according to referee comments,
to appear in Discrete Mathematic
Circuits through specified edges
AbstractWe prove a theorem implying the conjecture of Woodall [14] that, given any k independent edges in a (k+1)-connected graph, there is a circuit containing all of them. This implies the truth of a conjecture of Berge [1, p.214] and provides strong evidence to a conjecture of Lovász [8]
Long cycles in graphs with large degree sums and neighborhood unions
We present and prove several results concerning the length of longest cycles in 2-connected or 1-tough graphs with large degree sums. These results improve many known results on long cycles in these graphs. We also consider the sharpness of the results and discuss some possible strengthenings
Degree Sums, k-Factors and Hamilton Cycles in Graphs
https://digitalcommons.memphis.edu/speccoll-faudreerj/1189/thumbnail.jp
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