Subdivisions of graphs: A generalization of paths and cycles

Abstract

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 cycle C 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 ⩾ 3 k and minimum degree ⩾ 2 k contains k disjoint cycles. Again, this may be restated as: every graph of order ⩾ 3 k and minimum degree ⩾ 2 k contains a subdivision of kK 3 . 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.