# Packing seagulls in graphs with no stable set of size three

# Packing seagulls in graphs with no stable set of size three

Hadwiger's conjecture is a well known open problem in graph theory. It states that every graph with chromatic number $k$, contains a certain structure, called a "clique minor" of size $k$. An interesting special case of the conjecture, that is still wide open, is when the graph $G$ does not contain three pairwise non-adjacent vertices. In this case, it should be true that $G$ contains a clique minor of size $t$ where $t\geq |V(G)|/2$. This remains open, but Jonah Blasiak proved it in the subcase when $|V(G)|$ is even and the vertex set of $G$ is the union of three cliques. Here we prove a strengthening of Blasiak's result: that the conjecture holds if some clique in $G$ contains at least $|V(G)|/4$ vertices.This is a consequence of a result about packing "seagulls''. A seagull in $G$ is an induced three-vertex path. It is not known in general how to decide in polynomial time whether a graph contains $k$ pairwise disjoint seagulls; but we answer this for graphs with no stable sets of size three.

This is joint work with Paul Seymour.