Very Well-Covered Graphs with the Erd\H{o}s-Ko-Rado Property
| Autor: | De Silva, Jessica, Dionne, Adam B., Dunkelberg, Aidan, Harris, Pamela E. |
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| Rok vydání: | 2021 |
| Předmět: | |
| Zdroj: | Involve 16 (2023) 35-47 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.2140/involve.2023.16.35 |
| Popis: | A family of independent $r$-sets of a graph $G$ is an $r$-star if every set in the family contains some fixed vertex $v$. A graph is $r$-EKR if the maximum size of an intersecting family of independent $r$-sets is the size of an $r$-star. Holroyd and Talbot conjecture that a graph is $r$-EKR as long as $1\leq r\leq\frac{\mu(G)}{2}$, where $\mu(G)$ is the minimum size of a maximal independent set. It is suspected that the smallest counterexample to this conjecture is a well-covered graph. Here we consider the class of very well-covered graphs $G^*$ obtained by appending a single pendant edge to each vertex of $G$. We prove that the pendant complete graph $K_n^*$ is $r$-EKR when $n \geq 2r$ and strictly so when $n>2r$. Pendant path graphs $P_n^*$ are also explored and the vertex whose $r$-star is of maximum size is determined. Comment: 10 pages |
| Databáze: | arXiv |
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