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

Abstract

A 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 universal labeling number of a graph G is the minimum number k such that G has universal labeling from $\{1,2,\ldots , k\}$ denoted it by $\overrightarrow{\chi _{u}}(G)$. We have $2\Delta (G)-2 \le \overrightarrow{\chi _{u}} (G)\le 2^{\Delta (G)}$, 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, $\overrightarrow{\chi _{u}} (G)\le f(\Delta (G))$?”. 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 ${{\mathbf {N}}}{{\mathbf {P}}}$-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.