Cordiality of Digraphs

TL;DR

This paper extends the concept of cordial labelings to directed graphs, focusing on a specific type called (2,3)-cordiality, and characterizes which digraphs like tournaments, wheels, and fans possess this property.

Contribution

It introduces and investigates (2,3)-cordiality for digraphs, providing classifications for tournaments, wheels, and fans.

Findings

Characterization of (2,3)-cordial tournaments

Identification of (2,3)-cordial orientations of wheels

Determination of (2,3)-cordial orientations of fans

Abstract

A $(0,1)$-labelling of a set is said to be {\em friendly} if approximately one half the elements of the set are labelled 0 and one half labelled 1. Let $g$ be a labelling of the edge set of a graph that is induced by a labelling $f$ of the vertex set. If both $g$ and $f$ are friendly then $g$ is said to be a {\em cordial} labelling of the graph. We extend this concept to directed graphs and investigate the cordiality of sets of directed graphs. We investigate a specific type of cordiality on digraphs, a restriction of quasigroup-cordiality called $(2,3)$-cordiality. A directed graph is $(2,3)$-cordial if there is a friendly labelling $f$ of the vertex set which induces a $(1,-1,0)$-labelling of the arc set $g$ such that about one third of the arcs are labelled 1, about one third labelled -1 and about one third labelled 0. In particular we determine which tournaments are $(2,3)$-cordial, which orientations of the $n$-wheel are $(2,3)$-cordial, and which orientations of the $n -$fan are $(2,3)$-cordial.