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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2402.02507 |
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| _version_ | 1866914666346708992 |
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| author | Srisawat, Supakorn Vichitkunakorn, Panupong |
| author_facet | Srisawat, Supakorn Vichitkunakorn, Panupong |
| contents | The $δ$-complement $G_δ$ of a graph $G$, introduced in 2022 by Pai et al., is a variant of the graph complement, where two vertices are adjacent in $G_δ$ if and only if they are of the same degree but not adjacent in $G$ or they are of different degrees but adjacent in $G$. In this paper, we provide the Nordhaus-Gaddum-type bounds, in the spirit of Nordhaus and Gaddum (1956), over the maximum degrees, the minimum degrees, the vertex connectivities, and the edge connectivities of a graph and its $δ$-complement. All bounds are attained except for the upper bounds on the product between the minimum degrees of a graph and its $δ$-complement, the vertex connectivities of a graph and its $δ$-complement, and the edge connectivities of a graph and its $δ$-complement. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_02507 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Degrees and Connectivities of a Graph and Its $δ$-Complement Srisawat, Supakorn Vichitkunakorn, Panupong Combinatorics The $δ$-complement $G_δ$ of a graph $G$, introduced in 2022 by Pai et al., is a variant of the graph complement, where two vertices are adjacent in $G_δ$ if and only if they are of the same degree but not adjacent in $G$ or they are of different degrees but adjacent in $G$. In this paper, we provide the Nordhaus-Gaddum-type bounds, in the spirit of Nordhaus and Gaddum (1956), over the maximum degrees, the minimum degrees, the vertex connectivities, and the edge connectivities of a graph and its $δ$-complement. All bounds are attained except for the upper bounds on the product between the minimum degrees of a graph and its $δ$-complement, the vertex connectivities of a graph and its $δ$-complement, and the edge connectivities of a graph and its $δ$-complement. |
| title | Degrees and Connectivities of a Graph and Its $δ$-Complement |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2402.02507 |