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Is the following notion of graph labeling (let me call it special labeling) ever studied in the literature?
A special labelling of a (simple undirected) graph $G$ is a function $f:V(G)\to\mathbb{N}$ such that $f(u)\neq f(v)$ whenever $u$ and $v$ have a common neighbour (i.e., $N(u)\cap N(v)\neq \emptyset$).

The closest I could find in the literature is strong subcoloring introduced in [1]. A strong subcoloring of $G$ is a function $f:V(G)\to\mathbb{N}$ such that $f(u)\neq f(v)$ whenever $d(u,v)=2$ (i.e. distance between $u$ and $v$ in $G$ is two). The difference between the two notions is that special labelling does not allow two adjacent vertices to have the same label if they have a common neighbour. Both notions are the same for triangle-free graphs.

Reference
[1] Shalu, M. A.; Vijayakumar, S.; Yamini, S. Devi; Sandhya, T. P., On the algorithmic aspects of strong subcoloring, J. Comb. Optim. 35, No. 4, 1312-1329 (2018). ZBL1400.90266.

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`Special labelling' is not exactly $L(0,1)$-coloring, but is very close. In $L(0,1)$-coloring, neighboring vertices can get the same colour even if they have a common neighbor. Speciall labelling do not allow this. Special labelling is already studied in the literature under the name injective coloring.

An injective colouring of a graph $G$ is a colouring $c$ of the vertices of $G$ that assigns different colours to any pair of vertices that have a common neighbour.

There are a number of papers on injective coloring. The main graph classes studied are planar graphs [1,2,3], chordal graphs [4], regular graphs [5], and sparse graphs [6]

References

[1] Bu, Yuehua; Qi, Chentao; Zhu, Junlei; Xu, Ting, Injective coloring of planar graphs, ZBL07300884.

[2] Lužar, Borut; Škrekovski, Riste; Tancer, Martin, Injective colorings of planar graphs with few colors, Discrete Math. 309, No. 18, 5636-5649 (2009). ZBL1209.05093.

[3] Bu, Yuehua; Wang, Chao; Yang, Sheng, List injective coloring of planar graphs., ZBL06986943.

[4] Hell, Pavol; Raspaud, André; Stacho, Juraj, On injective colourings of chordal graphs, Laber, Eduardo Sany (ed.) et al., LATIN 2008 Berlin: Springer (ISBN 978-3-540-78772-3/pbk). LNCS 4957, 520-530 (2008). ZBL1136.68463.

[5] Hahn, Geňa; Kratochvíl, Jan; Širáň, Jozef; Sotteau, Dominique, On the injective chromatic number of graphs, Discrete Math. 256, No. 1-2, 179-192 (2002). ZBL1007.05046.

[6] Cranston, Daniel W.; Kim, Seog-Jin; Yu, Gexin, Injective colorings of sparse graphs, Discrete Math. 310, No. 21, 2965-2973 (2010). ZBL1209.05075.

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$L(p,q)$ coloring is a function V $\rightarrow \mathbb{N}$ such that the labels on vertices at distance 1 differ by at least $p$ and vertices at distance 2 differ by at least $q$. Special labelling looks like $L(0,1)$ coloring.

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  • $\begingroup$ Are you aware of any paper that give non-trivial resuts on $L(0,1)$-coloring? $\endgroup$ Mar 22 at 8:42

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