On the neighbour sum distinguishing index of planar graphs

TL;DR

This paper proves that for large enough maximum degree, planar graphs can be properly edge coloured with at most one more colour than the maximum degree, ensuring adjacent vertices have distinct incident colour sums.

Contribution

It confirms a conjecture that planar graphs with sufficiently large maximum degree can be coloured to distinguish vertex sums using only  maximum degree plus one colours.

Findings

Validates the conjecture for planar graphs with maximum degree at least 28.

Shows that  maximum degree + 1 colours suffice for the sum-distinguishing edge colouring.

Provides a stronger result that the number of colours from Vizing's theorem is sufficient.

Abstract

Let $c$ be a proper edge colouring of a graph $G=(V,E)$ with integers $1,2,\ldots,k$. Then $k\geq \Delta(G)$, while by Vizing's theorem, no more than $k=\Delta(G)+1$ is necessary for constructing such $c$. On the course of investigating irregularities in graphs, it has been moreover conjectured that only slightly larger $k$, i.e., $k=\Delta(G)+2$ enables enforcing additional strong feature of $c$, namely that it attributes distinct sums of incident colours to adjacent vertices in $G$ if only this graph has no isolated edges and is not isomorphic to $C_5$. We prove the conjecture is valid for planar graphs of sufficiently large maximum degree. In fact even stronger statement holds, as the necessary number of colours stemming from the result of Vizing is proved to be sufficient for this family of graphs. Specifically, our main result states that every planar graph $G$ of maximum degree at least $28$ which contains no isolated edges admits a proper edge colouring $c:E\to\{1,2,\ldots,\Delta(G)+1\}$ such that $\sum_{e\ni u}c(e)\neq \sum_{e\ni v}c(e)$ for every edge $uv$ of $G$.