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{\bf B. Y. Stodolsky}
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{\bf On Domination in 2-Connected Cubic Graphs}
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In 1996, Reed proved that the domination number, $\gamma(G)$, of every
$n$-vertex graph $G$ with minimum degree at least $3$ is at most
$3n/8$ and conjectured that $\gamma(H)\leq\lceil n/3\rceil$ for every
connected $3$-regular (cubic) $n$-vertex graph $H$. In [1] this
conjecture was disproved by presenting a connected cubic graph $G$ on
$60$ vertices with $\gamma(G)=21$ and a sequence
$\{G_k\}_{k=1}^{\infty}$ of connected cubic graphs with
$\lim_{k\to\infty}{\gamma(G_k)\over|V(G_k)|}
\geq{1\over3}+{1\over69}$. All the counter-examples,
however, had cut-edges. On the other hand, in [2] it was
proved that $\gamma(G)\leq\ 4n/11$ for every connected cubic $n$-vertex
graph $G$ with at least $10$ vertices. In this note we construct a
sequence of graphs $\{G_k\}_{k=1}^{\infty}$ of $2$-connected cubic
graphs with $\lim_{k\to\infty}{\gamma(G_k)\over|V(G_k)|}
\geq{1\over3}+{1\over78}$, and a sequence
$\{G_l'\}_{l=1}^{\infty}$ of connected cubic graphs where for each
$G_l'$ we have ${\gamma(G_l')\over|V(G_l')|}
>{1\over3}+{1\over69}$.

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