Non-Hamiltonian 3-Regular Graphs with Arbitrary Girth

Haythorpe, Michael

doi:10.13189/ujam.2014.020111

Cited by 2 publications

(2 citation statements)

References 9 publications

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“…The catalog [48] introduced in this paper suggests that the answer to open problem 2 might be yes, but it is not currently known whether this is true in general. As pointed out earlier, the research of Haythorpe 2014 [42] suggests that non-Hamiltonian cubic graph of minimum order is always larger than a Hamiltonian cubic graph of minimum order. It is indeed likely that this might be true in general.…”

Section: Open Problemsmentioning

confidence: 87%

A new approach to catalog small graphs of high even girth

Nittoor¹

2016

Preprint

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A catalog of a class of (3, g) graphs for even girth g is introduced in this paper. A (k, g) graph is a regular graph with degree k and girth g. This catalog of (3, g) graphs for even girth g satisfying 6 ≤ g ≤ 16, has the following properties. Firstly, this catalog contains the smallest known (3, g) graphs. An appropriate class of cubic graphs for this catalog has been identified, such that the (3, g) graph of minimum order within the class is also the smallest known (3, g) graph. Secondly, this catalog contains (3, g) graphs for more orders than other listings. Thirdly, the class of graphs have been defined so that a practical algorithm to generate graphs can be created. Fourthly, this catalog is infinite, since the results are extended into knowledge about infinitely many graphs. The findings are as follows. Firstly, Hamiltonian bipartite graphs have been identified as a promising class of cubic graphs that can lead to a catalog of (3, g) graphs for even girth g with graphs for more orders than other listings, that is also expected to contain a (3, g) graph with minimum order. Secondly, this catalog of (3, g) graphs contains many non-vertex-transitive graphs. Thirdly, in order to make the computation more tractable, and at the same time, to enable deeper analysis on the results, symmetry factor has been introduced as a measure of the extent of rotational symmetry along the identified Hamiltonian cycle. The D3 chord index notation is introduced as a concise notation for cubic Hamiltonian bipartite graphs. The D3 chord index notation is twice as compact as the LCF notation, which is known as a concise notation for cubic Hamiltonian graphs. The D3 chord index notation can specify an infinite family of graphs. Fourthly, results on the minimum order for existence of a (3, g) Hamiltonian bipartite graph, and minimum value of symmetry factor for existence of a (3, g) Hamiltonian bipartite graph are of wider interest from an extremal graph theory perspective.

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Section: Open Problemsmentioning

confidence: 87%

A new approach to catalog small graphs of high even girth

Nittoor¹

2016

Preprint

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“…Indeed, it is not even known whether there exist general non-Hamiltonian, 3-connected, cubic graphs of high cyclic connectivity. The existence of such graphs is conjectured in [6] while according to [13], Thomassen conjectured the contrary (in personal communication in 1991). The 28-vertex Coxeter graph has cyclic connectivity 7, witnessing the largest known cyclic connectivity among non-Hamiltonian cubic 3-connected graphs.…”

Section: Properties Of Minimum Counterexamplesmentioning

confidence: 99%

Cuts in matchings of 3-connected cubic graphs

Knauer

Valicov

2019

European Journal of Combinatorics

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We discuss conjectures on Hamiltonicity in cubic graphs (Tait, Barnette, Tutte), on the dichromatic number of planar oriented graphs (Neumann-Lara), and on even graphs in digraphs whose contraction is strongly connected (Hochstättler). We show that all of them fit into the same framework related to cuts in matchings. This allows us to find a counterexample to the conjecture of Hochstättler and show that the conjecture of Neumann-Lara holds for all planar graphs on at most 26 vertices. Finally, we state a new conjecture on bipartite cubic oriented graphs, that naturally arises in this setting.

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Non-Hamiltonian 3-Regular Graphs with Arbitrary Girth

Cited by 2 publications

References 9 publications

A new approach to catalog small graphs of high even girth

A new approach to catalog small graphs of high even girth

Cuts in matchings of 3-connected cubic graphs

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