On the neighbour sum distinguishing index of planar graphs

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 $G$.Comment: 22 page

Progress on the adjacent vertex distinguishing edge colouring conjecture

A proper edge colouring of a graph is adjacent vertex distinguishing if no two adjacent vertices see the same set of colours. Using a clever application of the Local Lemma, Hatami (2005) proved that every graph with maximum degree $\Delta$ and no isolated edge has an adjacent vertex distinguishing edge colouring with $\Delta + 300$ colours, provided $\Delta$ is large enough. We show that this bound can be reduced to $\Delta + 19$. This is motivated by the conjecture of Zhang, Liu, and Wang (2002) that $\Delta + 2$ colours are enough for $\Delta \geq 3$.Comment: v2: Revised following referees' comment