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

TL;DR

This paper introduces new characterizations of bipartite and 3-colorable graphs using directional labeling of edges and directional adjacency matrices, providing alternative methods to identify graph colorability.

Contribution

It presents novel characterizations of bipartite and 3-colorable graphs through directional edge labeling and directional adjacency matrices, expanding existing graph coloring theory.

Findings

Characterizations of bipartite graphs via directional labeling.

Characterizations of 3-colorable graphs using directional matrices.

New methods for identifying graph colorability.

Abstract

A \emph{directional labeling} of an edge $\emph{uv}$ 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 $\ell(uv)=ab$, and $\ell(vu)=ba$. 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.