Probabilistic Coloring of Bipartite and Split Graphs

Abstract

We revisit in this paper the probabilistic coloring problem (probabilistic coloring) and focus ourselves on bipartite and split graphs. We first give some general properties dealing with the optimal solution. We then show that the unique 2-coloring achieves approximation ratio 2 in bipartite graphs under any system of vertex-probabilities and propose a polynomial algorithm achieving tight approximation ratio 8/7 under identical vertex-probabilities. Then we deal with restricted cases of bipartite graphs. Main results for these cases are the following. Under non-identical vertex-probabilities probabilistic coloring is polynomial for stars, for trees with bounded degree and a fixed number of distinct vertex-probabilities, and, consequently, also for paths with a fixed number of distinct vertex-probabilities. Under identical vertex-probabilities, probabilistic coloring is polynomial for paths, for even and odd cycles and for trees whose leaves are either at even or at odd levels. Next, we deal with split graphs and show that probabilistic coloring is NP-hard, even under identical vertex-probabilities. Finally, we study approximation in split graphs and provide a 2-approximation algorithm for the case of distinct probabilities and a polynomial time approximation schema under identical vertex-probabilities.