Parity Labeling in Signed Graphs

TL;DR

This paper introduces the concept of parity labeling in signed graphs, defines the 'rna' number, and characterizes certain classes of parity signed graphs, opening new research directions in graph theory.

Contribution

It initiates the study of parity labeling in signed graphs, defines the 'rna' number, and characterizes some classes of parity signed graphs.

Findings

Defined the 'rna' number for signed graphs

Characterized certain classes of parity signed graphs

Proposed directions for future research

Abstract

Let $S=(G, \sigma)$ be a signed graph where $G=(V, E)$ is a graph called the underlying graph of $S$ and $\sigma:E(G) \rightarrow \{+,~-\}$. Let $f:V(G) \rightarrow \{1,2,\dots,|V(G)|\}$ such that $\sigma(uv)=+$ if and only if $f(u)$ and $f(v)$ are of same parity and $\sigma(uv)=-$ if and only if $f(u)$ and $f(v)$ are of opposite parity. Under $f$ we get a signed graph $G_f$ denoted as $S$, which is a parity signed graph. In this paper, we initiate the study of parity labeling in signed graphs and we define and find `rna' number denoted as $\sigma^-(S)$ for some classes of signed graphs. We also characterize some signed graphs which are parity signed graphs. Some directions for further research are also suggested.