On the In-Out-Proper Orientations of Graphs

TL;DR

This paper introduces the concept of in-out-proper orientations in graphs, defining a new orientation number based on in-out-degree differences that differ for adjacent vertices.

Contribution

It formalizes the in-out-proper orientation number and explores its properties, providing new insights into graph orientations with degree constraints.

Findings

Defined the in-out-proper orientation number $ar{igchi}(G)$

Established bounds and properties of $ar{igchi}(G)$

Analyzed the in-out-proper orientations for various classes of graphs

Abstract

An orientation of a graph $G$ is {\it in-out-proper} if any two adjacent vertices have different in-out-degrees, where the in-out-degree of each vertex is equal to the in-degree minus the out-degree of that vertex. The {\it in-out-proper orientation number} of a graph $G$, denoted by $\overleftrightarrow{\chi}(G)$, is $ \min_{D\in \Gamma}\max_{v\in V(G)} |d_D^{\pm}(v)|$, where $\Gamma$ is the set of in-out-proper orientations of $G$ and $d_D^{\pm}(v)$ is the in-out-degree of the vertex $v$ in the orientation $D$.