On endomorphisms of alternating forms graph

A graph G is a pseudo-core if every endomorphism of G is either an automorphism or a colouring. Let F-q be the finite field with q elements and let Alt(m, q)(m >= 4) be the alternating forms graph on the vector space F-q(m). We prove that Alt(m, q) is a pseudo-core. Moreover, if m is odd, then Alt(m, q) is a core. If both m and q are even, then Alt(m, q) is not a core. (C) 2014 Elsevier B.V. All rights reserved.