## Abstract

For an integer \(d \ge 2\), an \(L(d\),1)*-labeling* of a graph \(G\) is a function \(f\) from its vertex set to the non-negative integers such that \({\vert }f(x) - f(y){\vert } \ge d\) if vertices \(x\) and \(y\) are adjacent, and \({\vert }f(x) - f(y){\vert } \ge \) 1 if \(x\) and \(y\) are at distance two. The minimum span over all the L(\(d\),1)-labelings of \(G\) is denoted by \(\lambda _{d}(G)\). For a given integer \(k \ge 2\), the *edge-path-replacement of*
\(G\) or \(G(P_{k})\) is the graph obtained from \(G\) by replacing each edge with a path \(P_{k}\) on \(k\) vertices. We show that the edges of \(G\) can be colored with \(\lceil \varDelta (G)/2\rceil \) colors so that each monochromatic subgraph has maximum degree at most 2 and use this fact to establish general upper bounds on \(\lambda _{d}(G(P_{k}))\) for \(k \ge 4\). As a corollary, we settle the following conjecture by Lü (J Comb Optim, 2012): for any graph \(G\) with \(\varDelta (G) \ge \) 2, \(\lambda _{2}(G(P_{4})) \le \varDelta (G)\) + 2. Moreover, \(\lambda _{2}(G(P_{4})) = \varDelta (G) + 1\) when \(\varDelta (G)\) is even and different from 2. We also show that the class of graphs \(G(P_{k})\) with \(k \ge \) 4 satisfies a conjecture by Havet and Yu (2008 Discrete Math 308:498–513) in the related area of (\(d,1\))-total labeling of graphs.

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## Acknowledgments

The authors would like to thank Sarah Spence Adams for participating in the initial brainstorming sessions and Paul Booth for contributing to an earlier version of this work. The authors would also like to thank the referees for their helpful comments and suggestions.

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Karst, N., Oehrlein, J., Troxell, D.S. *et al.* L(\(d\),1)-labelings of the edge-path-replacement by factorization of graphs.
*J Comb Optim* **30, **34–41 (2015). https://doi.org/10.1007/s10878-013-9632-x

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### Keywords

- L(2\(, \)1)-labeling
- L(\(d, \)1)-labeling
- (\(d, \)1)-Total labeling
- Edge-path-replacement
- Factorization of graphs