A characterization of regular partial cubes whose all convex cycles have the same lengths
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 2 n $2n$ where n ⩾ 4 $n\geqslant 4$...
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| Published in | Journal of graph theory Vol. 107; no. 3; pp. 550 - 558 |
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| Main Authors | , , |
| Format | Journal Article |
| Language | English |
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| ISSN | 0364-9024 1097-0118 |
| DOI | 10.1002/jgt.23126 |
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| 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
2
n $2n$ where
n
⩾
4 $n\geqslant 4$) if and only if all its convex cycles are 4‐cycles (resp., 6‐cycles,
2
n $2n$‐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—regular median graphs are hypercubes. |
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| AbstractList | 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 where ) if and only if all its convex cycles are 4‐cycles (resp., 6‐cycles, ‐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—regular median graphs are hypercubes. 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 2 n $2n$ where n ⩾ 4 $n\geqslant 4$) if and only if all its convex cycles are 4‐cycles (resp., 6‐cycles, 2 n $2n$‐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—regular median graphs are hypercubes. 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 $2n$ where n⩾4 $n\geqslant 4$) if and only if all its convex cycles are 4‐cycles (resp., 6‐cycles, 2n $2n$‐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—regular median graphs are hypercubes. |
| Author | Xie, Yan‐Ting Xu, Shou‐Jun Feng, Yong‐De |
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| SubjectTerms | almost‐median graphs convex cycles Cubes Graphs Hypercubes partial cubes regularity |
| Title | A characterization of regular partial cubes whose all convex cycles have the same lengths |
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