Diameter Constraints in 2-distance Graphs
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866914271419432960 |
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| author | Al-saadi, Oleksiy Natal, Joseph |
| author_facet | Al-saadi, Oleksiy Natal, Joseph |
| contents | For any finite, simple graph $G = (V,E)$, its $2$-distance graph $G_2$ is a graph having the same vertex set $V$ where two vertices are adjacent if and only if their distance is $2$ in $G$. Connectivity and diameter properties of these graphs have been well studied. For example, it has been shown that if ${\rm diam}(G) = k \geq 3$ then $\lceil \frac{1}{2} k \rceil \leq {\rm diam}(G_2)$, and that this bound is sharp. In this paper, we prove that ${\rm diam}(G_2) = \infty$ (that is, $G_2$ is disconnected) or otherwise ${\rm diam}(G_2) \leq k + 2$. In addition, we show that this inequality is sharp for any even $k$, a result that we verify for some higher orders through judicious use of a \textsc{sat} solver. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_01575 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Diameter Constraints in 2-distance Graphs Al-saadi, Oleksiy Natal, Joseph Combinatorics 68R10 G.2.2 For any finite, simple graph $G = (V,E)$, its $2$-distance graph $G_2$ is a graph having the same vertex set $V$ where two vertices are adjacent if and only if their distance is $2$ in $G$. Connectivity and diameter properties of these graphs have been well studied. For example, it has been shown that if ${\rm diam}(G) = k \geq 3$ then $\lceil \frac{1}{2} k \rceil \leq {\rm diam}(G_2)$, and that this bound is sharp. In this paper, we prove that ${\rm diam}(G_2) = \infty$ (that is, $G_2$ is disconnected) or otherwise ${\rm diam}(G_2) \leq k + 2$. In addition, we show that this inequality is sharp for any even $k$, a result that we verify for some higher orders through judicious use of a \textsc{sat} solver. |
| title | Diameter Constraints in 2-distance Graphs |
| topic | Combinatorics 68R10 G.2.2 |
| url | https://arxiv.org/abs/2501.01575 |