Deciding the On-line Chromatic Number of a Graph with Pre-Coloring is PSPACE-Complete

Online Chromatic Number is PSPACE-Complete

“…The paper shows that the problem is coNP-hard and lies in PSPACE. Later [13] he proved that if some part of the graph is precolored, i.e., some vertices are assigned some colors prior to the coloring game between Drawer and Painter and Drawer also reveals edges to the precolored vertices for each incoming vertex, then deciding whether χ O (G) ≤ k is PSPACE-complete. We call this decision problem Online Chromatic Number with Precoloring.…”

“…We call this decision problem Online Chromatic Number with Precoloring. The paper [13] conjectures that Online Chromatic Number (with no precolored part) is PSPACE-complete too. Interestingly, it is possible to decide χ O (G) ≤ 3 in polynomial time [7].…”

“…Inspired by [13], we prove the PSPACE-hardness of Online Chromatic Number by a reduction from Q3DNF-SAT, i.e., the satisfiability of a fully quantified formula in the 3disjunctive normal form (3-DNF). An example of such a formula is…”

“…We note that if we would give Painter some advantage like parallel edges, precolored vertices (as in [13]) or something that helps Painter distinguish different parts of the graph, the proof of PSPACE-hardness would be much simpler. However, our theorem holds for simple graphs without any such help.…”

“…One of the main open problems in the area is the computational complexity of deciding whether χ O (G) ≤ k for a specified simple graph G, given G and k on input; see e.g. Kudahl [13].…”