Ore-type conditions for existence of a jellyfish in a graph

The famous Dirac's Theorem states that for each n≥ 3 every n-vertex graph G with minimum degree δ(G)≥ n/2 has a hamiltonian cycle. When δ(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 δ(G)≥ (n-1)/3 contains a spanning 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≥ 56 every connected n-vertex graph G with δ(G)≥ (n-2)/3 contains a spanning 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 δ(G)≥ (n-2)/3 and some additional properties contains a spanning 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) ≥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.