17,969 research outputs found
Hamiltonian Strongly Regular Graphs
We give a sufficient condition for a distance-regular graph to be Hamiltonian. In particular, the Petersen graph is the only connected non-Hamiltonian strongly regular graph on fewer than 99 vertices.Distance-regular graphs;Hamilton cycles JEL-code
Minimal chordal sense of direction and circulant graphs
A sense of direction is an edge labeling on graphs that follows a globally
consistent scheme and is known to considerably reduce the complexity of several
distributed problems. In this paper, we study a particular instance of sense of
direction, called a chordal sense of direction (CSD). In special, we identify
the class of k-regular graphs that admit a CSD with exactly k labels (a minimal
CSD). We prove that connected graphs in this class are Hamiltonian and that the
class is equivalent to that of circulant graphs, presenting an efficient
(polynomial-time) way of recognizing it when the graphs' degree k is fixed
A Study of Sufficient Conditions for Hamiltonian Cycles
A graph G is Hamiltonian if it has a spanning cycle. The problem of determining if a graph is Hamiltonian is well known to be NP-complete. While there are several necessary conditions for Hamiltonicity, the search continues for sufficient conditions. In their paper, On Smallest Non-Hamiltonian Regular Tough Graphs (Congressus Numerantium 70), Bauer, Broersma, and Veldman stated, without a formal proof, that all 4-regular, 2-connected, 1-tough graphs on fewer than 18 nodes are Hamiltonian. They also demonstrated that this result is best possible. Following a brief survey of some sufficient conditions for Hamiltonicity, Bauer, Broersma, and Veldman\u27s result is demonstrated to be true for graphs on fewer than 16 nodes. Possible approaches for the proof of the n=16 and n=17 cases also will be discussed
Hamiltonian cycles and 1-factors in 5-regular graphs
It is proven that for any integer and ,
there exist infinitely many 5-regular graphs of genus containing a
1-factorisation with exactly pairs of 1-factors that are perfect, i.e. form
a hamiltonian cycle. For , this settles a problem of Kotzig from 1964.
Motivated by Kotzig and Labelle's "marriage" operation, we discuss two gluing
techniques aimed at producing graphs of high cyclic edge-connectivity. We prove
that there exist infinitely many planar 5-connected 5-regular graphs in which
every 1-factorisation has zero perfect pairs. On the other hand, by the Four
Colour Theorem and a result of Brinkmann and the first author, every planar
4-connected 5-regular graph satisfying a condition on its hamiltonian cycles
has a linear number of 1-factorisations each containing at least one perfect
pair. We also prove that every planar 5-connected 5-regular graph satisfying a
stronger condition contains a 1-factorisation with at most nine perfect pairs,
whence, every such graph admitting a 1-factorisation with ten perfect pairs has
at least two edge-Kempe equivalence classes. The paper concludes with further
results on edge-Kempe equivalence classes in planar 5-regular graphs.Comment: 27 pages, 13 figures; corrected figure
Improved asymptotic upper bounds for the minimum number of pairwise distinct longest cycles in regular graphs
We study how few pairwise distinct longest cycles a regular graph can have
under additional constraints. For each integer , we give exponential
improvements for the best asymptotic upper bounds for this invariant under the
additional constraint that the graphs are -regular hamiltonian graphs.
Earlier work showed that a conjecture by Haythorpe on a lower bound for this
invariant is false because of an incorrect constant factor, whereas our results
imply that the conjecture is even asymptotically incorrect. Motivated by a
question of Zamfirescu and work of Chia and Thomassen, we also study this
invariant for non-hamiltonian 2-connected -regular graphs and show that in
this case the invariant can be bounded from above by a constant for all large
enough graphs, even for graphs with arbitrarily large girth.Comment: Submitted for publicatio
A note on a conjecture concerning tree-partitioning 3-regular graphs
If G is a 4-connected maximal planar graph, then G is hamiltonian (by a theorem\ud
of Whitney), implying that its dual graph Gļæ½ is a cyclically 4-edge connected 3-\ud
regular planar graph admitting a partition of the vertex set into two parts, each\ud
inducing a tree in Gļæ½, a so-called tree-partition. It is a natural question whether\ud
each cyclically 4-edge connected 3-regular graph admits such a tree-partition.\ud
This was conjectured by Jaeger, and recently independently by the ļæ½rst author.\ud
The main result of this note shows that each connected 3-regular graph on n\ud
vertices admits a partition of the vertex set into two sets such that precisely\ud
12\ud
n+2 edges have end vertices in each set. This is a necessary condition for having\ud
a tree-partition. We also show that not all cyclically 3-edge connected 3-regular\ud
(planar) graphs admit a tree-partition, and present the smallest counterexample
Hamilton decompositions of 6-regular abelian Cayley graphs
In 1969, Lovasz asked whether every connected, vertex-transitive graph has a Hamilton path. This question has generated a considerable amount of interest, yet remains vastly open. To date, there exist no known connected, vertex-transitive graph that does not possess a Hamilton path. For the Cayley graphs, a subclass of vertex-transitive graphs, the following conjecture was made:
Weak LovƔsz Conjecture: Every nontrivial, finite, connected Cayley graph is hamiltonian.
The Chen-Quimpo Theorem proves that Cayley graphs on abelian groups flourish with Hamilton cycles, thus prompting Alspach to make the following conjecture:
Alspach Conjecture: Every 2k-regular, connected Cayley graph on a finite abelian group has a Hamilton decomposition.
Alspachās conjecture is true for k = 1 and 2, but even the case k = 3 is still open. It is this case that this thesis addresses.
Chapters 1ā3 give introductory material and past work on the conjecture. Chapter 3 investigates the relationship between 6-regular Cayley graphs and associated quotient graphs. A proof of Alspachās conjecture is given for the odd order case when k = 3. Chapter 4 provides a proof of the conjecture for even order graphs with 3-element connection sets that have an element generating a subgroup of index 2, and having a linear dependency among the other generators.
Chapter 5 shows that if Ī = Cay(A, {s1, s2, s3}) is a connected, 6-regular, abelian Cayley graph of even order, and for some1 ā¤ i ā¤ 3, Īi = Cay(A/(si), {sj1 , sj2}) is 4-regular, and Īi ā Cay(ā¤3, {1, 1}), then Ī has a Hamilton decomposition. Alternatively stated, if Ī = Cay(A, S) is a connected, 6-regular, abelian Cayley graph of even order, then Ī has a Hamilton decomposition if S has no involutions, and for some s ā S, Cay(A/(s), S) is 4-regular, and of order at least 4.
Finally, the Appendices give computational data resulting from C and MAGMA programs used to generate Hamilton decompositions of certain non-isomorphic Cayley graphs on low order abelian groups
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