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A characterization of regular partial cubes whose all convex cycles have the same lengths
Combinatorics · Submitted 28 Jan 2023
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim In this paper, we prove that a regular partial cube is a hypercube (resp., a Doubled Odd graph, an even cycle of length $2n$ where $n\geqslant 4$) if and only if all its convex cycles are 4-cycles (resp., 6-cycles, $2n$-cycles).
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The abstract, sentence by sentence
Partial cubes are graphs that can be isometrically embedded into hypercubes. Convex cycles play an important role in the study of partial cubes. In this paper, we prove that a regular partial cube is a hypercube (resp., a Doubled Odd graph, an even cycle of length $2n$ where $n\geqslant 4$) if and only if all its convex cycles are 4-cycles (resp., 6-cycles, $2n$-cycles). In particular, the partial cubes whose all convex cycles are 4-cycles are equivalent to almost-median graphs. Therefore, we conclude that regular almost-median graphs are exactly hypercubes, which generalizes the result by Mulder [J. Graph Theory, 4 (1980) 107--110] -- regular median graphs are hypercubes.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Partial cubes are graphs that can be isometrically embedded into hypercubes. | background 1.00 | 0.05 | 0.01 | asserted 1.00 |
| s1 | Convex cycles play an important role in the study of partial cubes. | background 1.00 | 0.04 | 0.01 | asserted 1.00 |
| s2 | In this paper, we prove that a regular partial cube is a hypercube (resp., a Doubled Odd graph, an even cycle of length $2n$ where $n\geqslant 4$) if and only if all its convex cycles are 4-cycles (resp., 6-cycles, $2n$-cycles). | result 0.97 | 0.95 | 0.92 | asserted 1.00 |
| s3 | In particular, the partial cubes whose all convex cycles are 4-cycles are equivalent to almost-median graphs. | interpretation 0.61 | 0.64 | 0.09 | asserted 1.00 |
| s4 | Therefore, we conclude that regular almost-median graphs are exactly hypercubes, which generalizes the result by Mulder [J. Graph Theory, 4 (1980) 107--110] -- regular median graphs are hypercubes. | interpretation 0.65 | 0.85 | 0.35 | asserted 1.00 |
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The paper
Original abstract
Partial cubes are graphs that can be isometrically embedded into hypercubes. Convex cycles play an important role in the study of partial cubes. In this paper, we prove that a regular partial cube is a hypercube (resp., a Doubled Odd graph, an even cycle of length $2n$ where $n\geqslant 4$) if and only if all its convex cycles are 4-cycles (resp., 6-cycles, $2n$-cycles). In particular, the partial cubes whose all convex cycles are 4-cycles are equivalent to almost-median graphs. Therefore, we conclude that regular almost-median graphs are exactly hypercubes, which generalizes the result by Mulder [J. Graph Theory, 4 (1980) 107--110] -- regular median graphs are hypercubes.
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