A graph is said to be k-colorable if it can be properly colored using k colors. For example, a bipartite graph is 2-colorable. To see this, just assign two different colors to the two disjoint sets in a bipartite graph.

How do you know if a graph is two colorable?

A graph is 2-colorable if we can color each of its vertices with one of two colors, say red and blue, in such a way that no two red vertices are connected by an edge, and no two blue vertices are connected by an edge (a k-colorable graph is defined in a similar way).

How do you know if a graph is three colorable?

Let x be a vertex in V (G) − (N[v] ∪ N2(v)). In any proper 3-coloring of G, if it exists, the vertex x either gets the same color as v or x receives a different color than v. Therefore it is enough to determine if any of the graphs G/xv and G ∪ xv are 3-colorable.