Dominating set based exact algorithms for 3-coloring

Abstract

We show that the 3-colorability problem can be solved in O ( 1.296 n ) time on any n-vertex graph with minimum degree at least 15. This algorithm is obtained by constructing a dominating set of the graph greedily, enumerating all possible 3-colorings of the dominating set, and then solving the resulting 2-list coloring instances in polynomial time. We also show that a 3-coloring can be obtained in 2 o ( n ) time for graphs having minimum degree at least ω ( n ) where ω ( n ) is any function which goes to ∞. We also show that if the lower bound on minimum degree is replaced by a constant (however large it may be), then neither a 2 o ( n ) time nor a 2 o ( m ) time algorithm is possible ( m denotes the number of edges) for 3-colorability unless Exponential Time Hypothesis (ETH) fails. We also describe an algorithm which obtains a 4-coloring of a 3-colorable graph in O ( 1.2535 n ) time.