Graphs Classes Related to Bipartite Graphs

We continue the study of the following general problem on vertex colorings of graphs. Suppose that some vertices of a graph G are assigned to some colors. Can this “precoloring” be extended to a proper coloring of G with at most k colors (for some given k)? Here we investigate the complexity status of precoloring extendibility on some graph classes which are related to bipartite graphs, giving a complete solution for graphs with “co-chromatic number” 2, i.e., admitting a partition V = V1∪V2 of the vertex set V such that each Vi induces a complete subgraph or an independent set. On one hand, we show that our problem is closely related to the bipartite maximum matching problem that leads to a polynomial solution for split graphs and for the complements of bipartite graphs. On the other hand, the problem turns out to be NP-complete on bipartite graphs.