Thue choosability of trees
A vertex colouring of a graph G G is nonrepetitive  if for any path P=(v1,v2,,v2r) P = ( v 1 , v 2 ,  , v 2 r ) in G G , the first half is coloured differently from the second half. The Thue choice number  of G G is the least integer ? ? such that for every ? ? -list assignment L L of G G , there exists a nonrepetitive L L -colouring of G G . We prove that for any positive integer ? ? , there is a tree T T with ? ch ( T ) > ? . On the other hand, it is proved that if G? G ? is a graph of maximum degree ? ? , and G G is obtained from G? G ? by attaching to each vertex v v of G? G ? a connected graph of tree-depth at most z z rooted at v v , then ? ch ( G ) ? c ( ? , z ) for some constant c(?,d) c ( ? , d ) depending only on ? ? and z z .
thue choosability tree thue sequence thue chromatic number non-repetitive colouring