The neighbour-sum-distinguishing edge-colouring game

Let :E(G)N=N{0} be an edge colouring of a graph G and :V(G)N the vertex colouring given by (v)=ev(e) for every vV(G). A neighbour-sum-distinguishing edge-colouring of G is an edge colouring such that for every edge uv in G, (u)(v). The neighbour-sum-distinguishing edge-colouring game on G is the 2-player game defined as follows. The two players, Alice and Bob, alternately colour an uncoloured edge of G. Alice wins the game if, when all edges are coloured, the so-obtained edge colouring is a neighbour-sum-distinguishing edge-colouring of G. Otherwise, Bob wins.In this paper we study the neighbour-sum-distinguishing edge-colouring game on various classes of graphs. In particular, we prove that Bob wins the game on the complete graph Kn, n3, whoever starts the game, except when n=4. In that case, Bob wins the game on K4 if and only if Alice starts the game.