Journal of Combinatorial Optimization

, Volume 31, Issue 2, pp 447–462

# $$L(2,1)$$-labeling for brick product graphs

Article

## Abstract

Let $$G=(V, E)$$ be a graph. Denote $$d_G(u, v)$$ the distance between two vertices $$u$$ and $$v$$ in $$G$$. An $$L(2, 1)$$-labeling of $$G$$ is a function $$f: V \rightarrow \{0,1,\cdots \}$$ such that for any two vertices $$u$$ and $$v$$, $$|f(u)-f(v)| \ge 2$$ if $$d_G(u, v) = 1$$ and $$|f(u)-f(v)| \ge 1$$ if $$d_G(u, v) = 2$$. The span of $$f$$ is the difference between the largest and the smallest number in $$f(V)$$. The $$\lambda$$-number of $$G$$, denoted $$\lambda (G)$$, is the minimum span over all $$L(2,1 )$$-labelings of $$G$$. In this article, we confirm Conjecture 6.1 stated in X. Li et al. (J Comb Optim 25:716–736, 2013) in the case when (i) $$\ell$$ is even, or (ii) $$\ell \ge 5$$ is odd and $$0 \le r \le 8$$.

## Keywords

L(2, 1)-labeling Brick product graph Graph labeling Frequency assignment problem

## Notes

### Acknowledgments

This work was supported by the National Natural Science Foundation of China under the grant 61309015 and National Science Council grant, NSC-98-2115-M-035-002-MY3.

## References

1. Bodlaender HL, Kloks T, Tan RB, van Leeuwen J (2004) The $$L$$(2, 1)-labeling problem on graphs. Comput J 47:193–204Google Scholar
2. Calamoneri T (2011) The $$L(h, k)$$-labeling problem: a updated aurvey and annotated bibliography. Comput J 54:1344–1371Google Scholar
3. Fiala J, Golovach PA, Kratochvíl JK (2005) Distance constrained labelings of graphs of bounded treewidth. In Proceedings of the 32th ICALP 9:360–372Google Scholar
4. Griggs JR, Yeh RK (1992) Labelling graphs with a condition at distance 2. SIAM J Discret Math 308:586–595
5. Hale WK (1980) Frequency assignment: theory and applications. Proc IEEE 68:1497–1514
6. Jha PK (2001) Optimal $$L$$(2, 1)-labeling of strong products of cycles. IEEE Trans Circ Sys I Fund Theory Appl 48:498–500Google Scholar
7. Klavžar S, Vesel A (2003) Computing graph invariants on rotagraphs using dynamic algorithm approach: the case of (2,1)-colorings and independence numbers. Discret Appl Math 129:449–460
8. Korz̆e D, Vesel A (2005) $$L$$(2, 1)-labeling of strong products of cycles. Inf Process Lett 94:183–190Google Scholar
9. Kratochvíl J, Kratsch D, Liedloff M (2007) Exact algorithms for $$L$$(2, 1)-labeling of graphs. MFCS 2007. LNCS 4708:513–524Google Scholar
10. Li X, Mak-Hau V, Zhou S (2013) The $$L(2, 1)$$-labelling problem for cubic cayley graphs on dihedral groups. J Comb Optim 25:716–736Google Scholar
11. Schwarz C, Troxell DS (2006) $$L(2, 1)$$-labelings of products of two cycles. Disc Appl Math 154:1522–1540Google Scholar
12. Whittlesey MA, Georges JP, Mauro DW (1995) On the $$\lambda$$ number of $$q_n$$ and related graphs. SIAM J Discret Math 8:499–506
13. Yeh RK (2006) A survey on labeling graphs with a condition at distance two. Discret Math 306:1217–1231