# Rank numbers for bent ladders

• Volume: 34, Issue: 2, page 309-329
• ISSN: 2083-5892

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## Abstract

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A ranking on a graph is an assignment of positive integers to its vertices such that any path between two vertices with the same label contains a vertex with a larger label. The rank number of a graph is the fewest number of labels that can be used in a ranking. The rank number of a graph is known for many families, including the ladder graph P2 × Pn. We consider how ”bending” a ladder affects the rank number. We prove that in certain cases the rank number does not change, and in others the rank number differs by only 1. We investigate the rank number of a ladder with an arbitrary number of bends

## How to cite

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Peter Richter, et al. "Rank numbers for bent ladders." Discussiones Mathematicae Graph Theory 34.2 (2014): 309-329. <http://eudml.org/doc/267567>.

@article{PeterRichter2014,
abstract = {A ranking on a graph is an assignment of positive integers to its vertices such that any path between two vertices with the same label contains a vertex with a larger label. The rank number of a graph is the fewest number of labels that can be used in a ranking. The rank number of a graph is known for many families, including the ladder graph P2 × Pn. We consider how ”bending” a ladder affects the rank number. We prove that in certain cases the rank number does not change, and in others the rank number differs by only 1. We investigate the rank number of a ladder with an arbitrary number of bends},
author = {Peter Richter, Emily Leven, Anh Tran, Bryan Ek, Jobby Jacob, Darren A. Narayan},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {graph colorings; rankings of graphs; rank number; Cartesian product of graphs; ladder graph; bent ladder graph},
language = {eng},
number = {2},
pages = {309-329},
title = {Rank numbers for bent ladders},
url = {http://eudml.org/doc/267567},
volume = {34},
year = {2014},
}

TY - JOUR
AU - Peter Richter
AU - Emily Leven
AU - Anh Tran
AU - Bryan Ek
AU - Jobby Jacob
AU - Darren A. Narayan
TI - Rank numbers for bent ladders
JO - Discussiones Mathematicae Graph Theory
PY - 2014
VL - 34
IS - 2
SP - 309
EP - 329
AB - A ranking on a graph is an assignment of positive integers to its vertices such that any path between two vertices with the same label contains a vertex with a larger label. The rank number of a graph is the fewest number of labels that can be used in a ranking. The rank number of a graph is known for many families, including the ladder graph P2 × Pn. We consider how ”bending” a ladder affects the rank number. We prove that in certain cases the rank number does not change, and in others the rank number differs by only 1. We investigate the rank number of a ladder with an arbitrary number of bends
LA - eng
KW - graph colorings; rankings of graphs; rank number; Cartesian product of graphs; ladder graph; bent ladder graph
UR - http://eudml.org/doc/267567
ER -

## References

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1. [1] H. Alpert, Rank numbers of grid graphs, Discrete Math. 310 (2010) 3324-3333. doi:10.1016/j.disc.2010.07.022[Crossref] Zbl1221.05280
2. [2] H.L. Bodlaender, J.S. Deogun, K. Jansen, T. Kloks, D. Kratsch, H. M¨uller and Zs. Tuza, Rankings of graphs, SIAM J. Discrete Math. 11 (1998) 168-181. doi:10.1137/S0895480195282550[Crossref]
3. [3] E. Bruoth and M. Horˇn´ak, Online-ranking numbers for cycles and paths, Discuss. Math. Graph Theory 19 (1999) 175-197. doi:10.7151/dmgt.1094[Crossref]
4. [4] C.-W. Chang, D. Kuo and H-C. Lin, Ranking numbers of graphs, Inform. Process. Lett. 110 (2010) 711-716. doi:10.1016/j.ipl.2010.05.025[Crossref]
5. [5] D. Dereniowski, Rank coloring of graphs, in: Graph Colorings, M. Kubale (Ed.), Contemp. Math. AMS 352 (2004) 79-93. doi:10.1090/conm/352/06[Crossref]
6. [6] J. Ghoshal, R. Laskar, and D. Pillone, Minimal rankings, Networks 28 (1996) 45-53. doi:10.1002/(SICI)1097-0037(199608)28:1h45::AID-NET6i3.0.CO;2-D[Crossref] Zbl0863.05071
7. [7] A.V. Iyer, H.D. Ratliff and G. Vijayan, Optimal node ranking of trees, Inform. Process. Lett. 28 (1988) 225-229. doi:10.1016/0020-0190(88)90194-9[Crossref] Zbl0661.68063
8. [8] R.E. Jamison, Coloring parameters associated with the rankings of graphs, Congr. Numer. 164 (2003) 111-127. Zbl1043.05049
9. [9] M. Katchalski, W. McCuaig and S. Seager, Ordered colourings, Discrete Math. 142 (1995) 141-154. doi:10.1016/0012-365X(93)E0216-Q[Crossref] Zbl0842.05032
10. [10] T. Kloks, H. M¨uller and C.K. Wong, Vertex ranking of asteroidal triple-free graphs, Inform. Process. Lett. 68 (1998) 201-206. doi:10.1016/S0020-0190(98)00162-8[Crossref]
11. [11] C.E. Leiserson, Area efficient graph layouts for VLSI, Proc. 21st Ann. IEEE Symposium, FOCS (1980) 270-281.
12. [12] S. Novotny, J. Ortiz, and D.A. Narayan, Minimal k-rankings and the rank number of P2 n, Inform. Process. Lett. 109 (2009) 193-198. doi:10.1016/j.ipl.2008.10.004[WoS][Crossref]
13. [13] J. Ortiz, H. King, A. Zemke and D.A. Narayan, Minimal k-rankings for prism graphs, Involve 3 (2010) 183-190. doi:10.2140/involve.2010.3.183[Crossref] Zbl1221.05159
14. [14] A. Sen, H. Deng and S. Guha, On a graph partition problem with application to VLSI Layout , Inform. Process. Lett. 43 (1992) 87-94. doi:10.1016/0020-0190(92)90017-P [Crossref] Zbl0764.68132

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