## 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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## References

Adams SS, Booth P, Jaffe H, Zinnen SL (2012) Exact \(\lambda \)-numbers of generalized Petersen graphs of certain higher orders and on Möbius strips. Discret Appl Math 160:436–447

Calamoneri T (2011) The L(h, k)-labelling problem: a survey and annotated bibliography. Comput J 54:1344–1371

Cerioli MR, Posner DFD (2012) On L(2,1)-coloring split, chordal bipartite and weakly chordal graphs. Discret Appl Math 160:2655–2661

Charpentier C, Montassier M, Raspauld A (2012) L(p, q)-labeling of sparse graphs. J Comb Optim. doi:10.1007/s10878-012-9507-6

Chia ML, Kuo D, Yan JH, Yang SR (2012) (p, q)-total labeling of complete graphs. J Comb Optim. doi:10.1007/s10878-012-9471-1

Georges JP, Mauro DW, Whittlesey MA (1994) Relating path coverings to vertex labelings with a condition at distance two. Discret Math 135:103–111

Georges JP, Mauro DW (1995) Generalized vertex labeling with a condition at distance two. Congr Numer 109:141–159

Gonçalves D (2008) On the L(p,1)-labelling of graphs. Discret Math 308:1405–1414

Griggs JR, Yeh RK (1992) Labeling graphs with a condition at distance two. SIAM J Discret Math 5:585–595

Havet F, Reed B, Sereni JS- (2012) Griggs and Yeh’s conjecture and L(p,1)-labelings. SIAM J Discret Math 26:145–168

Havet F, Yu ML (2008) (p, 1)-Total labeling of graphs. Discret Math 308:498–513

Lin W, Wu J (2012) Distance two edge labelings of lattices. J Comb Optim. doi:10.1007/s10878-012-9508-5

Lü D (2012) L(2,1)-labelings of the edge-path-replacement of a graph. J Comb Optim. doi:10.1007/s10878-012-9470-2

Lü D, Lin N (2012) L(d,1)-labelings of the edge-path-replacement of a graph. J Comb Optim. doi:10.1007/s10878-012-9487-6

Panda BS, Goel P (2012) L(2, 1)-labeling of dually chordal graphs and strongly orderable graphs. Info Proc Lett 112:552–556

Petersen J (1891) Die theorie der regularen graphen. Acta Math 15:193–220

Wang F, Lin W (2012) Group path covering and L(j, k)-labelings of diameter two graphs. Info Proc Lett 112:124–128

Wu Q, Shiu WC, Sun PK (2012) Circular L(j, k)-labeling number of direct product of path and cycle. J Comb Optim. doi:10.1007/s10878-012-9520-9

Yeh RK (2006) A survey on labeling graphs with a condition at distance two. Discret Math 306:1217–1231

Zhai M, Lu C, Shu J (2012) A note on L(2, 1)-labelling of trees. Acta Mathematicae Applicatae Sinica 28:395–400

## 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