$Z_2\times Z_2$-cordial cycle-free hypergraphs

TL;DR

This paper investigates $Z_2 imes Z_2$-cordial labelings in hypergraphs, proving that all $p$-uniform hypertrees with $p>2$ and certain path hypergraphs are $Z_2 imes Z_2$-cordial, unlike some trees.

Contribution

The authors establish that $p$-uniform hypertrees with $p>2$ and path hypergraphs with edges of size at least 3 are $Z_2 imes Z_2$-cordial, extending previous results on trees.

Findings

All $p$-uniform hypertrees for $p>2$ are $Z_2 imes Z_2$-cordial.

Path hypergraphs with edges of size at least 3 are $Z_2 imes Z_2$-cordial.

Not all hypergraphs of maximum degree 1 are $Z_2 imes Z_2$-cordial; a necessary and sufficient condition is provided.

Abstract

Hovey introduced $A$-cordial labelings as a generalization of cordial and harmonious labelings \cite{Hovey}. If $A$ is an Abelian group, then a labeling $f \colon V (G) \rightarrow A$ of the vertices of some graph $G$ induces an edge labeling on $G$, the edge $uv$ receives the label $f (u) + f (v)$. A graph $G$ is $A$-cordial if there is a vertex-labeling such that (1) the vertex label classes differ in size by at most one and (2) the induced edge label classes differ in size by at most one. The problem of $A$-cordial labelings of graphs can be naturally extended for hypergraphs. It was shown that not every $2$-uniform hypertree (i.e., tree) admits a $Z_2\times Z_2$-cordial labeling \cite{Pechnik}. The situation changes if we consider $p$-uniform hypetrees for a bigger $p$. We prove that a $p$-uniform hypertree is $Z_2\times Z_2$-cordial for any $p>2$, and so is every path hypergraph in which all edges have size at least~3. The property is not valid universally in the class of hypergraphs of maximum degree~1, for which we provide a necessary and sufficient condition.