On bipartization of cubic graphs by removal of an independent set

Abstract

We study a new problem for cubic graphs: bipartization of a cubic graph Q by deleting sufficiently large independent set I. It can be expressed as follows: Given an integerkand a connectedn-vertex tripartite cubic graphQ=(V,E)with independence numberα(Q), doesQcontain an independent setIof sizeksuch thatQ−Iis bipartite? We are interested for which values of k the answer to this question is affirmative. We prove constructively that if α(Q)≥2n/5, then the answer is positive for each k satisfying ⌊(n−α(Q))/2⌋≤k≤α(Q). It remains an open question if a similar construction is possible for α(Q)<2n/5.We also show that this problem with α(Q)≥2n/5 and k satisfying ⌊n/3⌋≤k≤α(Q) can be related to semi-equitable graph 3-coloring, where one color class is of size k, and the subgraph induced by the remaining vertices is equitably 2-colored. This means that Q has a coloring of type (k,⌈(n−k)/2⌉,⌊(n−k)/2⌋).