  The network coloring game has been proposed in the literature of social
sciences as a model for conflict-resolution circumstances. The players of the
game are the vertices of a graph with $n$ vertices and maximum degree $\Delta$.
The game is played over rounds, and in each round all players simultaneously
choose a color from a set of available colors. Players have local information
of the graph: they only observe the colors chosen by their neighbors and do not
communicate or cooperate with one another. A player is happy when she has
chosen a color that is different from the colors chosen by her neighbors,
otherwise she is unhappy, and a configuration of colors for which all players
are happy is a proper coloring of the graph. It has been shown in the
literature that, when the players adopt a particular greedy randomized
strategy, the game reaches a proper coloring of the graph within $O(\log(n))$
rounds, with high probability, provided the number of colors available to each
player is at least $\Delta+2$. In this note we show that a modification of the
aforementioned greedy strategy yields likewise a proper coloring of the graph,
provided the number of colors available to each player is at least $\Delta+1$,
and results in a simple randomized distributed algorithm for the
$(\Delta+1)$-coloring problem..
