Even the Easiest(?) Graph Coloring Problem is not Easy in Streaming!

We study a graph coloring problem that is otherwise easy but becomes quite non-trivial in the one-pass streaming model. In contrast to previous graph coloring problems in streaming that try to find an assignment of colors to vertices, our main work is on estimating the number of conflicting or monochromatic edges given a coloring function that is streaming along with the graph; we call the problem {\sc Conflict-Est}. The coloring function on a vertex can be read or accessed only when the vertex is revealed in the stream. If we need the color on a vertex that has streamed past, then that color, along with its vertex, has to be stored explicitly. We provide algorithms for a graph that is streaming in different variants of the one-pass vertex arrival streaming model, viz. the {\sc Vertex Arrival} ({\sc VA}), {Vertex Arrival With Degree Oracle} ({\sc VAdeg}), {\sc Vertex Arrival in Random Order} ({\sc VArand}) models, with special focus on the random order model. We also provide matching lower bounds for most of the cases. The mainstay of our work is in showing that the properties of a random order stream can be exploited to design streaming algorithms for estimating the number of conflicting edges. We have also obtained a lower bound, though not matching the upper bound, for the random order model. Among all the three models vis-a-vis this problem, we can show a clear separation of power in favor of the {\sc VArand} model.