Subdivisions of graphs: A generalization of paths and cycles

One of the basic results in graph theory is Dirac’s theorem, that every graph of order n 3 and minimum degree n/2 is Hamiltonian. This may be restated as: if a graph of order n and minimum degree n/2 contains a cycleC then it contains a spanning cycle, which is just a spanning subdivision of C. We show that the same conclusion is true if instead of C, we choose any graph H such that every connected component of H is non-trivial and contains at most one cycle. The degree bound can be improved to (n− t)/2 if H has t components that are trees. We attempt a similar generalization of the Corrádi–Hajnal theorem that every graph of order 3k and minimum degree 2k contains k disjoint cycles. Again, this may be restated as: every graph of order 3k andminimum degree 2k contains a subdivision of kK3. We show that if H is any graph of order n with k components, each of which is a cycle or a non-trivial tree, then every graph of order n and minimum degree n− k contains a subdivision of H.