Stable cuts, NAC-colourings and flexible realisations of graphs
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| Format: | Preprint |
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2024
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| _version_ | 1866914529077624832 |
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| author | Clinch, Katie Garamvölgyi, Dániel Haslegrave, John Huynh, Tony Legerský, Jan Nixon, Anthony |
| author_facet | Clinch, Katie Garamvölgyi, Dániel Haslegrave, John Huynh, Tony Legerský, Jan Nixon, Anthony |
| contents | A (2-dimensional) realisation of a graph $G$ is a pair $(G,p)$, where $p$ maps the vertices of $G$ to $\mathbb{R}^2$. A realisation is flexible if it can be continuously deformed while keeping the edge lengths fixed, and rigid otherwise. We say that $G$ is rigid if every generic realisation of $G$ is rigid; otherwise, $G$ is flexible.
In this paper, we investigate the relationship between stable cuts and graphs which are either flexible, or admit a flexible (not necessarily generic) realisation with positive edge lengths. We strengthen a result of Chen and Yu, who proved that every $n$-vertex graph with at most $2n-4$ edges has a stable cut, by showing that every flexible graph has a stable cut. The existence of a stable cut is a sufficient, but not necessary, condition for a flexible realisation to exist. Using a result of Le and Pfender on stable cuts, we prove a conjecture of Grasegger, Legerský and Schicho that characterises the minimally rigid graphs which admit a flexible realisation.
Additionally, we investigate the number of NAC-colourings in various graphs. A NAC-colouring is a type of edge colouring introduced by Grasegger, Legerský and Schicho, who showed that the existence of such a colouring characterises the existence of a flexible realisation with positive edge lengths. We provide an upper bound on the number of NAC-colourings for arbitrary graphs, and construct families of graphs, including rigid and minimally rigid ones, for which this number is exponential in the number of vertices. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_16018 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Stable cuts, NAC-colourings and flexible realisations of graphs Clinch, Katie Garamvölgyi, Dániel Haslegrave, John Huynh, Tony Legerský, Jan Nixon, Anthony Combinatorics 52C25, 05C15 A (2-dimensional) realisation of a graph $G$ is a pair $(G,p)$, where $p$ maps the vertices of $G$ to $\mathbb{R}^2$. A realisation is flexible if it can be continuously deformed while keeping the edge lengths fixed, and rigid otherwise. We say that $G$ is rigid if every generic realisation of $G$ is rigid; otherwise, $G$ is flexible. In this paper, we investigate the relationship between stable cuts and graphs which are either flexible, or admit a flexible (not necessarily generic) realisation with positive edge lengths. We strengthen a result of Chen and Yu, who proved that every $n$-vertex graph with at most $2n-4$ edges has a stable cut, by showing that every flexible graph has a stable cut. The existence of a stable cut is a sufficient, but not necessary, condition for a flexible realisation to exist. Using a result of Le and Pfender on stable cuts, we prove a conjecture of Grasegger, Legerský and Schicho that characterises the minimally rigid graphs which admit a flexible realisation. Additionally, we investigate the number of NAC-colourings in various graphs. A NAC-colouring is a type of edge colouring introduced by Grasegger, Legerský and Schicho, who showed that the existence of such a colouring characterises the existence of a flexible realisation with positive edge lengths. We provide an upper bound on the number of NAC-colourings for arbitrary graphs, and construct families of graphs, including rigid and minimally rigid ones, for which this number is exponential in the number of vertices. |
| title | Stable cuts, NAC-colourings and flexible realisations of graphs |
| topic | Combinatorics 52C25, 05C15 |
| url | https://arxiv.org/abs/2412.16018 |