KAIST Discrete Math Seminar

A split of a graph is a partition (A,B) of the vertex set V(G) having subsets A₀ of A and B₀ of B such that |A|,|B| > 1 and a vertex a in A is adjacent to a vertex b in B if and only if a is in A₀ and b is in B₀. A graph is prime (with respect to the split decomposition) if it has no split.

We prove that for each n, there exists N such that every prime graph on at least N vertices contains a vertex-minor isomorphic to either a cycle of length n or a graph consisting of two disjoint cliques of size n joined by a matching.

In this talk, we plan to describe a main tool, which is called a blocking sequence in a prime graph, and we will describe two big steps of the proof. And we will pose some open problems behind this result.

This is a joint work with Sang-il Oum.