On the total versions of 1-2-3-conjecture for graphs and hypergraphs

TL;DR

This paper advances the understanding of the 1-2-3 conjecture and its total versions for graphs and hypergraphs, providing new results and confirming the conjecture for specific hypergraph families.

Contribution

It improves known results on total versions of the conjecture and confirms it for several well-known hypergraph classes.

Findings

Confirmed the conjecture for complete n-partite hypergraphs

Established the conjecture for paths, cycles, and theta hypergraphs

Characterized hypergraphs based on the total coloring parameter

Abstract

In 2004, Karo\'nski, \L uczak and Thomason proposed $1$-$2$-$3$-conjecture: For every nice graph $G$ there is an edge weighting function $ w:E(G)\rightarrow\{1,2,3\} $ such that the induced vertex coloring is proper. After that, the total versions of this conjecture were suggested in the literature and recently, Kalkowski et al. have generalized this conjecture to hypergraphs. In this paper, some previously known results on the total versions are improved. Moreover, an affirmative answer is given to the conjecture for some well-known families of hypergraphs like complete $n$-partite hypergraphs, paths, cycles, theta hypergraphs and some geometric planes. Also, these hypergraphs are characterized based on the corresponding parameter.