AVD-total-colouring of complete equipartite graphs ?
An AVD-total-colouring  of a simple graph G G is a mapping ?:V(G)?E(G)?C ? : V ( G ) ? E ( G ) ? C , C C a set of colours, such that: (i) for each pair of adjacent or incident elements x,y?V(G)?E(G) x , y ? V ( G ) ? E ( G ) , ?(x)??(y) ? ( x ) ? ? ( y ) ; (ii) for each pair of adjacent vertices x,y?V(G) x , y ? V ( G ) , sets {?(x)}?{?(xv):xv?E(G),v?V(G)} { ? ( x ) } ? { ? ( x v ) : x v ? E ( G ) , v ? V ( G ) } and {?(y)}?{?(yv):yv?E(G),v?V(G)} { ? ( y ) } ? { ? ( y v ) : y v ? E ( G ) , v ? V ( G ) } are distinct. The AVD-total-chromatic number , ? a ? ( G ) , is the smallest number of colours for which G G admits an AVD-total-colouring. In 2005, Zhang etal. conjectured that ? a ? ( G ) ? ? ( G ) + 3 for any simple graph G G . In this article this conjecture is verified for any complete equipartite graph. Moreover, if G G is a complete equipartite graph of even order, then it is shown that ? a ? ( G ) = ? ( G ) + 2 .
complete equipartite graph total-colouring adjacent-vertex-distinguishing-total-colouring