Characterizations of 2-Colorable (Bipartite) and 3-Colorable Graphs

Abstract

A directional labeling of an edge uv in a graph G = (V, E) by an ordered pair ab is a labeling of the edge uv such that the label on uv in the direction from u to v is ℓ(uv) = ab, and ℓ(vu) = ba. New characterizations of 2-colorable (bipartite) and 3-colorable graphs are obtained in terms of directional labeling of edges of a graph by ordered pairs ab and ba. In addition we obtain characterizations of 2-colorable and 3-colorable graphs in terms of matrices called directional adjacency matrices.