Matching Theory and Barnette's Conjecture

Maximilian Gorsky; Raphael Steiner; Sebastian Wiederrecht

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Matching Theory and Barnette's Conjecture

Maximilian Gorsky, Raphael Steiner, Sebastian Wiederrecht

Combinatorics · Submitted 23 Feb 2022

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim This allows us to relax the requirement of planarity to give the equivalent conjecture that all cubic, 3-connected, Pfaffian, bipartite graphs are Hamiltonian.

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What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.26 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.51 · mentions code or data availability 0.05.

The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; show every role.

Barnette's Conjecture claims that all cubic, 3-connected, planar, bipartite graphs are Hamiltonian. We give a translation of this conjecture into the matching-theoretic setting. This allows us to relax the requirement of planarity to give the equivalent conjecture that all cubic, 3-connected, Pfaffian, bipartite graphs are Hamiltonian. A graph, other than the path of length three, is a brace if it is bipartite and any two disjoint edges are part of a perfect matching. Our perspective allows us to observe that Barnette's Conjecture can be reduced to cubic, planar braces. We show a similar reduction to braces for cubic, 3-connected, bipartite graphs regarding four stronger versions of Hamiltonicity. Note that in these cases we do not need planarity. As a practical application of these results, we provide some supplements to a generation procedure for cubic, 3-connected, planar, bipartite graphs discovered by Holton et al. [Hamiltonian Cycles in Cubic 3-Connected Bipartite Planar Graphs, JCTB, 1985]. These allow us to check whether a graph we generated is a brace.

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What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Barnette's Conjecture claims that all cubic, 3-connected, planar, bipartite graphs are Hamiltonian.	background 1.00	0.04	0.06	asserted 0.58
s1	We give a translation of this conjecture into the matching-theoretic setting.	contribution 0.83	0.94	0.06	asserted 1.00
s2	This allows us to relax the requirement of planarity to give the equivalent conjecture that all cubic, 3-connected, Pfaffian, bipartite graphs are Hamiltonian.	result 0.50	0.88	0.33	asserted 0.98
s3	A graph, other than the path of length three, is a brace if it is bipartite and any two disjoint edges are part of a perfect matching.	background 1.00	0.10	0.03	asserted 1.00
s4	Our perspective allows us to observe that Barnette's Conjecture can be reduced to cubic, planar braces.	result 0.78	0.89	0.16	asserted 0.90
s5	We show a similar reduction to braces for cubic, 3-connected, bipartite graphs regarding four stronger versions of Hamiltonicity.	result 0.70	0.94	0.18	asserted 1.00
s6	Note that in these cases we do not need planarity.	interpretation 0.64	0.66	0.07	asserted 0.99
s7	As a practical application of these results, we provide some supplements to a generation procedure for cubic, 3-connected, planar, bipartite graphs discovered by Holton et al. [Hamiltonian Cycles in Cubic 3-Connected Bipartite Planar Graphs, JCTB, 1985].	contribution 0.94	0.88	0.05	asserted 1.00
s8	These allow us to check whether a graph we generated is a brace.	contribution 0.49	0.87	0.04	asserted 0.93

What kind of paper the classifier reports this is: theoretical result 0.96 · evidence from proof or derivation 0.97.

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The paper

Original abstract

Barnette's Conjecture claims that all cubic, 3-connected, planar, bipartite graphs are Hamiltonian. We give a translation of this conjecture into the matching-theoretic setting. This allows us to relax the requirement of planarity to give the equivalent conjecture that all cubic, 3-connected, Pfaffian, bipartite graphs are Hamiltonian. A graph, other than the path of length three, is a brace if it is bipartite and any two disjoint edges are part of a perfect matching. Our perspective allows us to observe that Barnette's Conjecture can be reduced to cubic, planar braces. We show a similar reduction to braces for cubic, 3-connected, bipartite graphs regarding four stronger versions of Hamiltonicity. Note that in these cases we do not need planarity. As a practical application of these results, we provide some supplements to a generation procedure for cubic, 3-connected, planar, bipartite graphs discovered by Holton et al. [Hamiltonian Cycles in Cubic 3-Connected Bipartite Planar Graphs, JCTB, 1985]. These allow us to check whether a graph we generated is a brace.

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