On the Advice Complexity of the Online L(2, 1)-Coloring Problem on Paths and Cycles

In an L(2,1)-coloring of a graph, the vertices are colored with colors from an ordered set such that neighboring vertices get colors that have distance at least 2 and vertices at distance 2 in the graph get different colors. We consider the problem of finding an L(2,1)-coloring using a minimum range of colors in an online setting where the vertices arrive in consecutive time steps together with information about their neighbors and vertices at distance two among the previously revealed vertices. For this, we restrict our attention to paths and cycles.