Updating the hamiltonian problem—A survey

Gould, Ronald J.

doi:10.1002/jgt.3190150204

Cited by 130 publications

(67 citation statements)

References 112 publications

(109 reference statements)

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Many such results involve degree conditions and other neighborhood conditions. Such results have been surveyed in several papers (see, e.g., [12,23,28]). We do not want to discuss such conditions in this survey, but here is a connectivity-only result.…”

Section: Related Results With Additional Conditionsmentioning

confidence: 99%

How Many Conjectures Can You Stand? A Survey

Broersma

Ryjáček

Vrána

2011

Graphs and Combinatorics

View full text Add to dashboard Cite

show abstract

Section: Related Results With Additional Conditionsmentioning

confidence: 99%

How Many Conjectures Can You Stand? A Survey

Broersma

Ryjáček

Vrána

2011

Graphs and Combinatorics

View full text Add to dashboard Cite

show abstract

“…. , n} from Section 1 and B (7,3) is the set {∅, {1}, {2}, {3}, {4}, {5}, {6}, {7}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 5}, {2, 6}, {2, 7}, {3, 6}, {3, 7}, {4, 7}, {1, 4, 7}}.…”

Section: A Generalizationmentioning

confidence: 99%

“…For surveys of the general topic of Gray codes, for many more examples of such codes in a variety of combinatorial families, and for pointers to recent literature in the subject we suggest [1,3,6,8]. …”

mentioning

confidence: 99%

Combinatorial families that are exponentially far from being listable in Gray code sequence

Chinburg

Savage

Wilf

1999

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. Let S(n) be a collection of subsets of {1, ..., n}. In this paper we study numerical obstructions to the existence of orderings of S(n) for which the cardinalities of successive subsets satisfy congruence conditions. Gray code orders provide an example of such orderings. We say that an ordering of S(n) is a Gray code order if successive subsets differ by the adjunction or deletion of a single element of {1, . . . , n}. The cardinalities of successive subsets in a Gray code order must alternate in parity. It follows that if d(S(n)) is the difference between the number of elements of S(n) having even (resp. odd) cardinality, then |d(S(n))| − 1 is a lower bound for the cardinality of the complement of any subset of S(n) which can be listed in Gray code order.For g ≥ 2, the collection B(n, g) of g-blockfree subsets of {1, . . . , n} is defined to be the set of all subsets S of {1, . . . , n} such that |a − b| ≥ g if a, b ∈ S and a = b. We will construct a Gray code order for B(n, 2). In contrast, for g > 2 we find the precise (positive) exponential growth rate of d(B(n, g)) with n as n → ∞. This implies B(n, g) is far from being listable in Gray code order if n is large. Analogous results for other kinds of orderings of subsets of B(n, g) are proved using generalizations of d (B(n, g)) . However, we will show that for all g, one can order B(n, g) so that successive elements differ by the adjunction and/or deletion of an integer from {1, . . . , n}.We show that, over an A-letter alphabet, the words of length n which contain no block of k consecutive letters cannot, in general, be listed so that successive words differ by a single letter. However, if k > 2 and A > 2 or if k = 2 and A > 3, such a listing is always possible.

show abstract

“…More recent material on hamiltonian cycles (and related concepts) in graphs and hypergraphs can be found in Gould's comprehensive three-part survey [48][49][50]. For an overview of results on hamiltonicity in directed graphs, see [90] by Kühn and Osthus.…”

Section: Introductionmentioning

confidence: 99%

On Hypohamiltonian and Almost Hypohamiltonian Graphs

Zamfirescu

2014

Journal of Graph Theory

View full text Add to dashboard Cite

A graph G is almost hypohamiltonian if G is non‐hamiltonian, there exists a vertex w such that G−w is non‐hamiltonian, and for any vertex v≠w the graph G−v is hamiltonian. We prove the existence of an almost hypohamiltonian graph with 17 vertices and of a planar such graph with 39 vertices. Moreover, we find a 4‐connected almost hypohamiltonian graph, while Thomassen's question whether 4‐connected hypohamiltonian graphs exist remains open. We construct planar almost hypohamiltonian graphs of order n for every n≥74. During our investigation we draw connections between hypotraceable, hypohamiltonian, and almost hypohamiltonian graphs, and discuss a natural extension of almost hypohamiltonicity. Finally, we give a short argument disproving a conjecture of Chvátal (originally disproved by Thomassen), strengthen a result of Araya and Wiener on cubic planar hypohamiltonian graphs, and mention open problems.

show abstract

Updating the hamiltonian problem—A survey

Cited by 130 publications

References 112 publications

How Many Conjectures Can You Stand? A Survey

How Many Conjectures Can You Stand? A Survey

Combinatorial families that are exponentially far from being listable in Gray code sequence

On Hypohamiltonian and Almost Hypohamiltonian Graphs

Contact Info

Product

Resources

About