On the order of uniquely (k,m)-colourable graphs

Abstract

For integers k⩾1 and m⩾2 a (k,m)-colouring of a graph G is a colouring of the vertices of G in k colours such that no m-clique of G is monocoloured. The mth chromatic number χ _m(G) of G is the least k for which Ghas a ( /IT>)-colouring. A graph G is uniquely (k,m)-colourable if χ_m(G)=k and any two (k,m)-colourings of G induce the same partition of V(G). We prove that, for k⩾2 and m⩾3, there exists a uniquely (k,m)-colourable graph of order n if and only if n⩾k(m−1)+m(k−1). In the process, we determine the only uniquely (2,m)-colourable graph of order 3m−2 and describe the structure of all the uniquely (k,m)-colourable graphs of order k(m−1)+m(k−1).