Tight bound for independent domination of cubic graphs without $4$-cycles
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2021
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| _version_ | 1866914646321004544 |
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| author | Cho, Eun-Kyung Choi, Ilkyoo Kwon, Hyemin Park, Boram |
| author_facet | Cho, Eun-Kyung Choi, Ilkyoo Kwon, Hyemin Park, Boram |
| contents | Given a graph $G$, a dominating set of $G$ is a set $S$ of vertices such that each vertex not in $S$ has a neighbor in $S$. The domination number of $G$, denoted $γ(G)$, is the minimum size of a dominating set of $G$. The independent domination number of $G$, denoted $i(G)$, is the minimum size of a dominating set of $G$ that is also independent.
Recently, Abrishami and Henning proved that if $G$ is a cubic graph with girth at least $6$, then $i(G) \le \frac{4}{11}|V(G)|$. We show a result that not only improves upon the upper bound of the aforementioned result, but also applies to a larger class of graphs, and is also tight. Namely, we prove that if $G$ is a cubic graph without $4$-cycles, then $i(G) \le \frac{5}{14}|V(G)|$, which is tight. Our result also implies that every cubic graph $G$ without $4$-cycles satisfies $\frac{i(G)}{γ(G)} \le \frac{5}{4}$, which partially answers a question by O and West in the affirmative. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2112_11720 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Tight bound for independent domination of cubic graphs without $4$-cycles Cho, Eun-Kyung Choi, Ilkyoo Kwon, Hyemin Park, Boram Combinatorics 05C69 Given a graph $G$, a dominating set of $G$ is a set $S$ of vertices such that each vertex not in $S$ has a neighbor in $S$. The domination number of $G$, denoted $γ(G)$, is the minimum size of a dominating set of $G$. The independent domination number of $G$, denoted $i(G)$, is the minimum size of a dominating set of $G$ that is also independent. Recently, Abrishami and Henning proved that if $G$ is a cubic graph with girth at least $6$, then $i(G) \le \frac{4}{11}|V(G)|$. We show a result that not only improves upon the upper bound of the aforementioned result, but also applies to a larger class of graphs, and is also tight. Namely, we prove that if $G$ is a cubic graph without $4$-cycles, then $i(G) \le \frac{5}{14}|V(G)|$, which is tight. Our result also implies that every cubic graph $G$ without $4$-cycles satisfies $\frac{i(G)}{γ(G)} \le \frac{5}{4}$, which partially answers a question by O and West in the affirmative. |
| title | Tight bound for independent domination of cubic graphs without $4$-cycles |
| topic | Combinatorics 05C69 |
| url | https://arxiv.org/abs/2112.11720 |