# Is there any polynomial upper bound for the universal labeling of graphs?

## TL;DR

This paper investigates whether a polynomial upper bound exists for the universal labeling number of graphs, providing bounds, specific results for trees and 3-regular graphs, and discussing computational complexity and probabilistic insights.

## Contribution

It introduces bounds on the universal labeling number, proves polynomial bounds for trees, characterizes 3-regular graphs with universal labelings, and explores complexity and probabilistic aspects.

## Key findings

- Universal labeling number for trees is O(Δ^3).
- Determining if a 3-regular graph has universal labeling number 4 is NP-complete.
- Proposes bounds and probabilistic approaches related to the universal labeling number.

## Abstract

A {\it universal labeling} of a graph $G$ is a labeling of the edge set in $G$ such that in every orientation $\ell$ of $G$ for every two adjacent vertices $v$ and $u$, the sum of incoming edges of $v$ and $u$ in the oriented graph are different from each other. The {\it universal labeling number} of a graph $G$ is the minimum number $k$ such that $G$ has {\it universal labeling} from $\{1,2,\ldots, k\}$ denoted it by $\overrightarrow{\chi_{u}}(G) $. We have $2\Delta(G)-2 \leq \overrightarrow{\chi_{u}} (G)\leq 2^{\Delta(G)}$, where $\Delta(G)$ denotes the maximum degree of $G$. In this work, we offer a provocative question that is:" Is there any polynomial function $f$ such that for every graph $G$, $\overrightarrow{\chi_{u}} (G)\leq f(\Delta(G))$?". Towards this question, we introduce some lower and upper bounds on their parameter of interest. Also, we prove that for every tree $T$, $\overrightarrow{\chi_{u}}(T)=\mathcal{O}(\Delta^3) $. Next, we show that for a given 3-regular graph $G$, the universal labeling number of $G$ is 4 if and only if $G$ belongs to Class 1. Therefore, for a given 3-regular graph $G$, it is an $ \mathbf{NP} $-complete to determine whether the universal labeling number of $G$ is 4. Finally, using probabilistic methods, we almost confirm a weaker version of the problem.