Saturation numbers for families of graph subdivisions


  • Research supported in part by UCD GK12 project, NSF award no. 0742434. The author acknowledges support of the National Science Foundation through a fellowship funded by the grant “EMSW21-MCTP: Research Experience for Graduate Students” (NSF DMS 08-38434).


For a family math formula of graphs, a graph G is math formula-saturated if G contains no member of math formula as a subgraph, but for any edge math formula in math formula, math formula contains some member of math formula as a subgraph. The minimum number of edges in an math formula-saturated graph of order n is denoted math formula. 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 math formula denote the family of H-subdivisions, including H itself. In this paper, we study math formula when H is one of math formula or math formula, obtaining several exact results and bounds. In particular, we determine math formula exactly for math formula and show for n sufficiently large that there exists a constant math formula such that math formula. For math formula we show that math formula will suffice, and that this can be improved slightly depending on the value of math formula. We also give an upper bound on math formula for all t and show that math formula. This provides an interesting contrast to a 1937 result of Wagner (Math Ann, 114 (1937), 570–590), who showed that edge-maximal graphs without a K5-minor have at least math formula edges.