On a vertex-capturing game

In this paper, we study the recently introduced scoring game played on graphs called Edge-Balanced Index Game. This is a graph by two players, Alice and Bob, who take turns colouring an uncoloured edge of graph. plays first colours edges red, while Bob blue. The ends once all have been coloured. A player captures vertex if more than half its incident are coloured that player, most vertices wins. Using classical arguments from field, prove general properties game. Namely, there no in which can win (if optimally), never capture 2 optimally). Through dedicated arguments, then investigate specific game, focus outcome when particular classes. Specifically, determine paths, cycles, complete bipartite graphs, Cartesian grids, give partial results for trees graphs.