The endomorphisms of Grassmann graphs

Abstract

A graph G is a core if every endomorphism of G is an automorphism. A graph is called a pseudo-core if every its endomorphism is either an automorphism or a colouring. Suppose that Jq(n, m) is a Grassmann graph over a finite field with q elements. We show that every Grassmann graph is a pseudo-core. Moreover, J2(4, 2) is not a core and Jq(2k + 1, 2) (k ≥ 2) is a core.