A conjecture on the prevalence of cubic bridge graphs

Filar, Jerzy A.; Haythorpe, Michael; Nguyen, Giang T.

doi:10.7151/dmgt.1485

Cited by 9 publications

(11 citation statements)

References 5 publications

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“…Future work on this topic might seek to address not only edge-connectivity, but cyclic edge-connectivity, that is, the size of the minimal edge-cutset that separates a graph into multiple components each containing a cycle. It is worth noting that the each of the (3,5,3), (3,6,3) and (3,7,3)-prisons have cyclic connectivity equal to their girth, leading to the second conjecture of this manuscript. It is easy to see that the non-Hamiltonian 3-regular, 3-connected graphs produced by the construction in Section 3 of this manuscript always have cyclic connectivity of 3.…”

Section: Small Prisonsmentioning

confidence: 83%

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Non-Hamiltonian 3-Regular Graphs with Arbitrary Girth

Haythorpe¹

2014

ujam

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It is well known that 3-regular graphs with arbitrarily large girth exist. Three constructions are given that use the former to produce non-Hamiltonian 3-regular graphs without reducing the girth, thereby proving that such graphs with arbitrarily large girth also exist. The resulting graphs can be 1-, 2-or 3-edge-connected depending on the construction chosen. From the constructions arise (naive) upper bounds on the size of the smallest non-Hamiltonian 3-regular graphs with particular girth. Several examples are given of the smallest such graphs for various choices of girth and connectedness.

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Section: Small Prisonsmentioning

confidence: 83%

“…It is well-known that graphs containing bridges (i.e. 1-edge-connected graphs) are always non-Hamiltonian; indeed, for 3-regular graphs it is conjectured that almost all non-Hamiltonian graphs contain bridges [6].…”

Section: Bridge Constructionmentioning

confidence: 99%

Non-Hamiltonian 3-Regular Graphs with Arbitrary Girth

Haythorpe¹

2014

ujam

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“…Future work on this topic might seek to address not only edge-connectivity, but cyclic edge-connectivity, that is, the size of the minimal edge-cutset that separates a graph into multiple components each containing a cycle. It is worth noting that the each of the (3, 5, 3), (3,6,3) and (3, 7, 3)-prisons have cyclic connectivity equal to their girth, leading to the second conjecture of this manuscript.…”

Section: Small Prisonsmentioning

confidence: 86%

Non-Hamiltonian 3-Regular Graphs with Arbitrary Girth

Haythorpe¹

2019

Preprint

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“…In Haythorpe [15], two of these polyhedra P in variables corresponding to arcs were constructed, it was conjectured (based on empirical evidence) that both polyhedra are empty for any cubic bridge graph. Bridge graphs are always non-Hamiltonian, and it was conjectured in [11] that, asymptotically, almost all non-Hamiltonian graphs are bridge graphs. However, no other non-Hamiltonian graphs were detected using either of the polyhedra in Haythorpe [15].…”

Section: Definition 12 a Gene Is A Connected Cubic Graph That Contamentioning

confidence: 99%

A new heuristic for detecting non-Hamiltonicity in cubic graphs

Filar

Haythorpe

Rossomakhine

2015

Computers & Operations Research

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We analyse a polyhedron which contains the convex hull of all Hamiltonian cycles of a given undirected connected cubic graph. Our constructed polyhedron is defined by polynomially-many linear constraints in polynomially-many continuous (relaxed) variables. Clearly, the emptiness of the constructed polyhedron implies that the graph is non-Hamiltonian. However, whenever a constructed polyhedron is non-empty, the result is inconclusive. Hence, the following natural question arises: if we assume that a non-empty polyhedron implies Hamiltonicity, how frequently is this diagnosis incorrect? We prove that, in the case of bridge graphs, the constructed polyhedron is always empty. We also demonstrate that some nonbridge non-Hamiltonian cubic graphs induce empty polyhedra as well. We compare our approach to the famous Dantzig-Fulkerson-Johnson relaxation of a TSP, and give empirical evidence which suggests that the latter is infeasible if and only if our constructed polyhedron is also empty. By considering special edge cut sets which are present in most cubic graphs, we describe a heuristic approach, built on our constructed polyhedron, for which incorrect diagnoses of non-Hamiltonian graphs as Hamiltonian appear to be very rare. In particular, for cubic graphs containing up to 18 vertices, only four out of 45,982 undirected connected cubic graphs were so misdiagnosed. By constrast, we demonstrate that an equivalent heuristic, when built on the Dantzig-Fulkerson-Johnson relaxation of a TSP, is mostly unsuccessful in identifying additional non-Hamiltonian graphs. These empirical results suggest that polynomial algorithms based on our constructed polyhedron may be able to correctly identify Hamiltonicity of a cubic graph in all but rare cases.

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A conjecture on the prevalence of cubic bridge graphs

Cited by 9 publications

References 5 publications

Non-Hamiltonian 3-Regular Graphs with Arbitrary Girth

Non-Hamiltonian 3-Regular Graphs with Arbitrary Girth

Non-Hamiltonian 3-Regular Graphs with Arbitrary Girth

A new heuristic for detecting non-Hamiltonicity in cubic graphs

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