Saturation numbers for families of graph subdivisions

For a family F of graphs, a graph G is F-saturated if G contains no member of F as a subgraph, but for any edge uv in G, G+ uv contains some member of F as a subgraph. The minimum number of edges in an F-saturated graph of order n is denoted sat(n,F). A subdivision of a graph H, or an H-subdivision, is a graph G obtained from H by replacing the edges of H with internally disjoint paths of arbitrary length. We let S(H) denote the family of H-subdivisions, including H itself. In this paper, we study sat(n, S(H)) when H is one of Ct or Kt, obtaining several exact results and bounds. In particular, we determine sat(n, S(Ct)) exactly for 3 ≤ t ≤ 5 and show for n sufficiently large that there exists a constant ct such that 5 4 n ≤ sat(n, S(Ct)) ≤ ( 5 4 + ct t ) n. For t ≥ 36 we show that ct = 8 will suffice, and that this can be improved slightly depending on the value of t (mod 8). We also give an upper bound on sat(n, S(Kt)) for all t and show that sat(n, S(K5)) = ⌈ 3n+4 2 ⌉. This provides an interesting contrast to a 1935 result of Wagner [23], who showed that edge-maximal graphs without a K5-minor have at least 11n 6 edges.