Three colorability characterized by shrinking of locally connected subgraphs into triangles

Abstract

We consider a recursive reduction of a graph where each 3-colorable locally connected subgraph is replaced by a triangle so that equally colored vertices coincide into one vertex. There is one-to-one correspondence between 3-colorings of the original graph and the graph resulting from the reduction. Using this approach we characterize classes of 3-colorable graphs and present polynomial time algorithms that recognize graphs from the classes and color them by three colors.