The 1/3-conjectures for domination in cubic graphs
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arXiv
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| Natura: | Preprint |
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2024
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| _version_ | 1866910631538458624 |
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| author | Dorbec, Paul Henning, Michael Antony |
| author_facet | Dorbec, Paul Henning, Michael Antony |
| contents | A set S of vertices in a graph G is a dominating set of G if every vertex not in S is adjacent to a vertex in S . The domination number of G, denoted by $γ$(G), is the minimum cardinality of a dominating set in G. In a breakthrough paper in 2008, L{ö}wenstein and Rautenbach proved that if G is a cubic graph of order n and girth at least 83, then $γ$(G) $\le$ n/3. A natural question is if this girth condition can be lowered. The question gave birth to two 1/3-conjectures for domination in cubic graphs. The first conjecture, posed by Verstraete in 2010, states that if G is a cubic graph on n vertices with girth at least 6, then $γ$(G) $\le$ n/3. The second conjecture, first posed as a question by Kostochka in 2009, states that if G is a cubic, bipartite graph of order n, then $γ$(G) $\le$n/3. In this paper, we prove Verstraete's conjecture when there is no 7-cycle and no 8-cycle, and we prove the Kostochka's related conjecture for bipartite graphs when there is no 4-cycle and no 8-cycle. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_17820 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The 1/3-conjectures for domination in cubic graphs Dorbec, Paul Henning, Michael Antony Discrete Mathematics Combinatorics A set S of vertices in a graph G is a dominating set of G if every vertex not in S is adjacent to a vertex in S . The domination number of G, denoted by $γ$(G), is the minimum cardinality of a dominating set in G. In a breakthrough paper in 2008, L{ö}wenstein and Rautenbach proved that if G is a cubic graph of order n and girth at least 83, then $γ$(G) $\le$ n/3. A natural question is if this girth condition can be lowered. The question gave birth to two 1/3-conjectures for domination in cubic graphs. The first conjecture, posed by Verstraete in 2010, states that if G is a cubic graph on n vertices with girth at least 6, then $γ$(G) $\le$ n/3. The second conjecture, first posed as a question by Kostochka in 2009, states that if G is a cubic, bipartite graph of order n, then $γ$(G) $\le$n/3. In this paper, we prove Verstraete's conjecture when there is no 7-cycle and no 8-cycle, and we prove the Kostochka's related conjecture for bipartite graphs when there is no 4-cycle and no 8-cycle. |
| title | The 1/3-conjectures for domination in cubic graphs |
| topic | Discrete Mathematics Combinatorics |
| url | https://arxiv.org/abs/2401.17820 |