  The famous Dirac's Theorem states that for each $n\geq 3$ every $n$-vertex
graph $G$ with minimum degree $\delta(G)\geq n/2$ has a hamiltonian cycle. When
$\delta(G)< n/2$, this cannot be guaranteed, but the existence of some other
specific subgraphs can be provided. Gargano, Hell, Stacho and Vaccaro proved
that every connected $n$-vertex graph $G$ with $\delta(G)\geq (n-1)/3$ contains
a spanning {\em spider}, i.e., a spanning tree with at most one vertex of
degree at least $3$. Later, Chen, Ferrara, Hu, Jacobson and Liu proved the
stronger (and exact) result that for $n\geq 56$ every connected $n$-vertex
graph $G$ with $\delta(G)\geq (n-2)/3$ contains a spanning {\em broom}, i.e., a
spanning spider obtained by joining the center of a star to an endpoint of a
path. They also showed that a $2$-connected graph $G$ with $\delta(G)\geq
(n-2)/3$ and some additional properties contains a spanning {\em jellyfish}
which is a graph obtained by gluing the center of a star to a vertex in a cycle
disjoint from that star. Note that every spanning jellyfish contains a spanning
broom.
  The goal of this paper is to prove an exact Ore-type bound which guarantees
the existence of a spanning jellyfish: We prove that if $G$ is a $2$-connected
graph on $n$ vertices such that every non-adjacent pair of vertices $(u,v)$
satisfies $d(u) + d(v) \geq \frac{2n-3}{3}$, then $G$ has a spanning jellyfish.
As corollaries, we obtain strengthenings of two results by Chen et al.: a
minimum degree condition guaranteeing the existence of a spanning jellyfish,
and an Ore-type sufficient condition for the existence of a spanning broom. The
corollaries are sharp for infinitely many $n$. One of the main ingredients of
our proof is a modification of the Hopping Lemma due to Woodall.
