Several Roman domination graph invariants on Kneser graphs
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arXiv
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866916122677215232 |
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| author | Zec, Tatjana Grbić, Milana |
| author_facet | Zec, Tatjana Grbić, Milana |
| contents | This paper considers the following three Roman domination graph invariants on Kneser graphs:
Roman domination, total Roman domination, and signed Roman domination.
For Kneser graph $K_{n,k}$, we present exact values for Roman domination number $γ_{R}(K_{n,k})$ and total Roman domination number $γ_{tR}(K_{n,k})$ proving that for $n\geqslant k(k+1)$, $γ_{R}(K_{n,k}) =γ_{tR}(K_{n,k}) = 2(k+1)$. For signed Roman domination number $γ_{sR}(K_{n,k})$, the new lower and upper bounds for $K_{n,2}$ are provided: we prove that for $n\geqslant 12$, the lower bound is equal to 2, while the upper bound depends on the parity of $n$ and is equal to 3 if $n$ is odd, and equal to $5$ if $n$ is even. For graphs of smaller dimensions, exact values are found by applying exact methods from literature. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2204_05664 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Several Roman domination graph invariants on Kneser graphs Zec, Tatjana Grbić, Milana Combinatorics This paper considers the following three Roman domination graph invariants on Kneser graphs: Roman domination, total Roman domination, and signed Roman domination. For Kneser graph $K_{n,k}$, we present exact values for Roman domination number $γ_{R}(K_{n,k})$ and total Roman domination number $γ_{tR}(K_{n,k})$ proving that for $n\geqslant k(k+1)$, $γ_{R}(K_{n,k}) =γ_{tR}(K_{n,k}) = 2(k+1)$. For signed Roman domination number $γ_{sR}(K_{n,k})$, the new lower and upper bounds for $K_{n,2}$ are provided: we prove that for $n\geqslant 12$, the lower bound is equal to 2, while the upper bound depends on the parity of $n$ and is equal to 3 if $n$ is odd, and equal to $5$ if $n$ is even. For graphs of smaller dimensions, exact values are found by applying exact methods from literature. |
| title | Several Roman domination graph invariants on Kneser graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2204.05664 |