Salvato in:
| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2016
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| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/1608.02107 |
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Sommario:
- For any graph $G$, we define the power $π(G)$ as the minimum of the largest number of neighbors in a $γ$-set of $G$, of any vertex, taken over all $γ$-sets of $G$. We show that $γ(G\square H)\geq \frac{π(G)}{2π(G) -1}γ(G)γ(H)$. Our methods allow us to prove the following statements for any graphs $G$ and $H$, (1) $γ(G\square H)\geq \frac{\lceil \frac{γ(G)}{2}\rceil}{2\lceil \frac{γ(G)}{2}\rceil-1}γ(G)γ(H)$ for odd $γ(G)$, (2) $γ(G\square H)\geq \frac{γ(G)}{2γ(G)-2}γ(G)γ(H)$, for even $γ(G)$, and (3) a short proof of Vizing's conjecture where $γ(G)=3$. Our argument relies on establishing efficient correspondences between dominating vertices and subsets of their neighborhoods and then showing a sufficient number of dominating vertices that horizontally dominate vertically undominated cells.