On Tuza's Conjecture in Dense Graphs
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866929348556095488 |
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| author | Chahua, Luis Gutierrez, Juan |
| author_facet | Chahua, Luis Gutierrez, Juan |
| contents | In 1982, Tuza conjectured that the size $τ(G)$ of a minimum set of edges that intersects every triangle of a graph $G$ is at most twice the size $ν(G)$ of a maximum set of edge-disjoint triangles of $G$. This conjecture was proved for several graph classes. In this paper, we present three results regarding Tuza's Conjecture for dense graphs. By using a probabilistic argument, Tuza proved its conjecture for graphs on $n$ vertices with minimum degree at least $\frac{7n}{8}$. We extend this technique to show that Tuza's conjecture is valid for split graphs with minimum degree at least $\frac{3n}{5}$; and that $τ(G) < \frac{28}{15}ν(G)$ for every tripartite graph with minimum degree more than $\frac{33n}{56}$. Finally, we show that $τ(G)\leq \frac{3}{2}ν(G)$ when $G$ is a complete 4-partite graph. Moreover, this bound is tight. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_11409 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Tuza's Conjecture in Dense Graphs Chahua, Luis Gutierrez, Juan Combinatorics Discrete Mathematics In 1982, Tuza conjectured that the size $τ(G)$ of a minimum set of edges that intersects every triangle of a graph $G$ is at most twice the size $ν(G)$ of a maximum set of edge-disjoint triangles of $G$. This conjecture was proved for several graph classes. In this paper, we present three results regarding Tuza's Conjecture for dense graphs. By using a probabilistic argument, Tuza proved its conjecture for graphs on $n$ vertices with minimum degree at least $\frac{7n}{8}$. We extend this technique to show that Tuza's conjecture is valid for split graphs with minimum degree at least $\frac{3n}{5}$; and that $τ(G) < \frac{28}{15}ν(G)$ for every tripartite graph with minimum degree more than $\frac{33n}{56}$. Finally, we show that $τ(G)\leq \frac{3}{2}ν(G)$ when $G$ is a complete 4-partite graph. Moreover, this bound is tight. |
| title | On Tuza's Conjecture in Dense Graphs |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2405.11409 |