Asymptotically optimal neighbour sum distinguishing total colourings of graphs

TL;DR

This paper proves that for large graphs, the minimum number of colours needed for a neighbour sum distinguishing total colouring approaches the maximum degree, supporting a long-standing conjecture with an asymptotic result.

Contribution

It establishes an asymptotic upper bound on the neighbour sum distinguishing total chromatic number, approaching the maximum degree, using probabilistic methods.

Findings

Asymptotic bound $oxed{ ext{(1+o(1))} imes ext{max degree}}$ for the neighbour sum distinguishing total chromatic number.

Supports the conjecture that $oxed{ ext{max degree}+3}$ colours suffice for all graphs.

Uses probabilistic techniques and biased random colour assignment based on attractors.

Abstract

Consider a simple graph $G=(V,E)$ of maximum degree $\Delta$ and its proper total colouring $c$ with the elements of the set $\{1,2,\ldots,k\}$. The colouring $c$ is said to be \emph{neighbour sum distinguishing} if for every pair of adjacent vertices $u$, $v$, we have $c(u)+\sum_{e\ni u}c(e)\neq c(v)+\sum_{e\ni v}c(e)$. The least integer $k$ for which it exists is denoted by $\chi"_{\sum}(G)$, hence $\chi"_{\sum}(G) \geq \Delta+1$. On the other hand, it has been daringly conjectured that just one more label than presumed in the famous Total Colouring Conjecture suffices to construct such total colouring $c$, i.e., that $\chi"_{\sum}(G) \leq \Delta+3$ for all graphs. We support this inequality by proving its asymptotic version, $\chi"_{\sum}(G) \leq (1+o(1))\Delta$. The major part of the construction confirming this relays on a random assignment of colours, where the choice for every edge is biased by so called attractors, randomly assigned to the vertices, and the probabilistic result of Molloy and Reed on the Total Colouring Conjecture itself.