The neighbour-sum-distinguishing edge-colouring game

Let γ : E(G) −→ N∗ be an edge colouring of a graph G and σγ : V (G) −→ N∗ the vertex colouring given by σγ(v) = ∑ e3v γ(e) for every v ∈ V (G). A neighbour-sumdistinguishing edge-colouring of G is an edge colouring γ such that for every edge uv in G, σγ(u) 6= σγ(v). The study of neighbour-sum-distinguishing edge-colouring of graphs was initiated by Karoński, Łuczak and Thomason [8]. They conjectured that every graph with no isolated edge admits a neighbour-sum-distinguishing edge-colouring with three colours. We consider a game version of neighbour-sum-distinguishing edge-colouring. The neighboursum-distinguishing edge-colouring game on a graphG is a 2-player game where the two players, called 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. Therefore, Bob’s goal is to produce an edge colouring such that two neighbouring vertices get the same sum, while Alice’s goal is to prevent him from doing so. The neighbour-sum-distinguishing edge-colouring game on G with Alice having the first move will be referred to as the A-game on G. The neighbour-sum-distinguishing edge-colouring game on G with Bob having the first move will be referred to as the B-game on G. 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, n ≥ 3, whoever starts the game, except when n = 4. In that case, Bob wins the game on K4 if and only if he starts the game.