Graph Cordiality -- Extremes and Preservers

TL;DR

This paper explores the properties of cordial and (2,3)-orientable graphs, providing examples, bounds, and characterizations of transformations that preserve these graph classes.

Contribution

It introduces the concepts of cordiality and (2,3)-orientability, presents examples of minimal non-graphs, and characterizes structure-preserving operators.

Findings

Identified smallest noncordial and non-(2,3)-orientable graphs.

Established upper bounds on edges for cordial and (2,3)-orientable graphs.

Proved that structure-preserving operators are vertex permutations.

Abstract

An undirected graph is said to be cordial if there is a friendly (0,1)-labeling of the vertices that induces a friendly (0,1)-labeling of the edges. An undirected graph $G$ is said to be $(2,3)$-orientable if there exists a friendly (0,1)-labeling of the vertices of $G$ such that about one third of the edges are incident to vertices labeled the same. That is, there is some digraph that is an orientation of $G$ that is $(2,3)$-cordial. Examples of the smallest noncordial/non-$(2,3)$-orientable graphs are given and upper bounds on the possible number of edges in a cordial/$(2,3)$-orientable graph are presented. It is also shown that if $T$ is a linear operator on the set of all undirected graphs on $n$ vertices that strongly preserves the set of cordial graphs or the set of $(2,3)$-orientable graphs then $T$ is a vertex permutation..