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Induced Saturation for P5P_5

Joint work with Marthe Bonamy, Carla Groenland, Tash Morrison and Alex Scott.

A graph GG is said to be induced saturated for HH if GG contains no induced copy of HH, but adding or removing any edge creates an induced copy. While this definition holds for any graph HH, research has so far focused on paths and, to a lesser extent, cycles.

Paths are nearly completely understood: the cases P2P_2 and P3P_3 are easily handled by the empty and complete graphs respectively, the case P4P_4 was shown to be impossible by Martin and Smith in 2012, and the case of PnP_n where n6n \geq 6 has recently been solved by Vojtěch Dvořák. This leaves just the case P5P_5 which we close here.

Armed with some code which reads in graphs from stdin and some which computes the number of paths of a given length, it didn’t take long to create a simple program to check if graphs are induced saturated for paths. In fact, the first version was cobbled together essentially in the time it took my collaborators to check the Petersen graph is induced saturated for P6P_6. To find an induced saturated graph for P5P_5 (and many other small paths), we first need a list of graphs to check. All small graphs is a reasonable starting point, and it yields two graphs for P6P_6, the Petersen graph and a graph on 11 vertices, but running this code for all graphs on 12 vertices is already infeasible. The next list we tried was all connected vertex-transitive graphs, which we could do up to 31 vertices. This provides plenty of examples for small graphs, including 5 for P5P_5.

The first two examples are pictured above. Only the second one seems to have any sort of special structure; it is the circulant graph on 19 vertices and jumps 1, 2, 3, 5, 7 and 8. I don’t know of any nice argument why these graphs are induced saturated for P5P_5, so checking that these work is left as an exercise for the reader.

Despite providing plenty of examples, checking all vertex transitive graphs didn’t reveal many obvious patterns, especially with no simple way of working out the structure of the examples (if any). The next thing to try was families of graphs. This gave us some families which are probably induced saturated such as the Kneser graphs K(n,2)K(n,2) for P6P_6 (as proved by Cho, Choi and Park) and the bipartite Kneser graphs H(n,2)H(n, 2) for P12P_{12}, but it was slow to come up with, generate and check each family of graphs. With a bit of effort, Mathematica can be made to output its list of graphs in Graph6 along with all the names associated with each graph. This offers a quick way to check many different families and is definitely something worth trying again in the future, even if it didn’t give what we were looking for this time.

We did find a couple of constructions which were induced saturated for infinitely many nn, both of which are cubic, but that’s for another time. Finally, by running our code for nearly 144 hours, we can say that the folded cube of order 7 isn’t PkP_k-induced saturated for any kk. Both of these results answer questions of Cho, Choi and Park.