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

A universal labeling of a graph G is a labeling of the edge set in G such that in every orientation l 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, . . . , k} denoted it by −→ χu(G). We have 2∆(G) − 2 ≤ −→ χu(G) ≤ 2 , where ∆(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, −→ χu(G) ≤ f(∆(G))?”. Towards this question, we introduce some lower and upper bounds on their parameter of interest. Also, we prove that for every tree T , −→ χu(T ) = O(∆ ). 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 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.