Deciding Relaxed Two-Colorability - A Hardness Jump

A coloring is proper if each color class induces connected components of order one (where the order of a graph is its number of vertices). Here we study relaxations of proper two-colorings, such that the order of the induced monochromatic components in one (or both) of the color classes is bounded by a constant. In a (C1, C2)-relaxed coloring of a graph G every monochromatic component induced by vertices of the first (second) color is of order at most C1 (C2, resp.). We are mostly concerned with (1, C)-relaxed colorings, in other words when/how is it possible to break up a graph into small components with the removal of an independent set. We prove that every graph of maximum degree at most three can be (1, 22)-relaxed colored and we give a quasilinear algorithm which constructs such a coloring. We also show that a similar statement cannot be true for graphs of maximum degree at most 4 in a very strong sense: we construct 4-regular graphs such that the removal of any independent set le...

• Type: Conference
• Year: 2006
• Where: ESA
• Authors: Robert Berke, Tibor Szabó
• Keywords: Algorithms, Connected Component, ESA 2006, Maximum Degree, Monochromatic Component