Cordiality of Digraphs

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.

Figures (8)

• Figure 1: A $(2,3)$-Cordial labellings of a 5-tournament
• Figure 2: $(2,3)$-Cordial labellings of 3- tournaments
• Figure 3: $(2,3)$-Cordial labellings of two 4-tournaments and two non (2,3) cordial 4-tournaments with their out-degree sequences.
• Figure 4: A 6-wheel graph.
• Figure 5: A 6-cycle-out-wheel graph.

Theorems & Definitions (22)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 3.1
• Definition 3.2
• Lemma 3.1
• Example 3.1
• Lemma 3.2
• Lemma 3.3