On the P3-hull numbers of q-Kneser graphs and Grassmann graphs

Let S be an n-dimensional vector space over the finite field Fq, where q is necessarily a prime power. Denote Kq(n,k) (resp. Jq(n,k)) to be the q-Kneser graph (resp. Grassmann graph) for k⩾1 whose vertices are the k-dimensional subspaces of S and two vertices v1 and v2 are adjacent if dim(v1∩v2)=0 (resp. dim(v1∩v2)=k−1). We consider the infection spreading in the q-Kneser graphs and the Grassmann graphs: a vertex gets infected if it has at least two infected neighbors. In this paper, we compute the P3-hull numbers of Kq(n,k) and Jq(n,k) respectively, which is the minimum size of a vertex set that eventually infects the whole graph.