# 2-Tone Colorings in Graph Products

Jennifer Loe; Danielle Middelbrooks; Ashley Morris; Kirsti Wash

Discussiones Mathematicae Graph Theory (2015)

- Volume: 35, Issue: 1, page 55-72
- ISSN: 2083-5892

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top## How to cite

topJennifer Loe, et al. "2-Tone Colorings in Graph Products." Discussiones Mathematicae Graph Theory 35.1 (2015): 55-72. <http://eudml.org/doc/271235>.

@article{JenniferLoe2015,

abstract = {A variation of graph coloring known as a t-tone k-coloring assigns a set of t colors to each vertex of a graph from the set \{1, . . . , k\}, where the sets of colors assigned to any two vertices distance d apart share fewer than d colors in common. The minimum integer k such that a graph G has a t- tone k-coloring is known as the t-tone chromatic number. We study the 2-tone chromatic number in three different graph products. In particular, given graphs G and H, we bound the 2-tone chromatic number for the direct product G×H, the Cartesian product G□H, and the strong product G⊠H.},

author = {Jennifer Loe, Danielle Middelbrooks, Ashley Morris, Kirsti Wash},

journal = {Discussiones Mathematicae Graph Theory},

keywords = {t-tone coloring; Cartesian product; direct product; strong product; -tone coloring; cartesian product},

language = {eng},

number = {1},

pages = {55-72},

title = {2-Tone Colorings in Graph Products},

url = {http://eudml.org/doc/271235},

volume = {35},

year = {2015},

}

TY - JOUR

AU - Jennifer Loe

AU - Danielle Middelbrooks

AU - Ashley Morris

AU - Kirsti Wash

TI - 2-Tone Colorings in Graph Products

JO - Discussiones Mathematicae Graph Theory

PY - 2015

VL - 35

IS - 1

SP - 55

EP - 72

AB - A variation of graph coloring known as a t-tone k-coloring assigns a set of t colors to each vertex of a graph from the set {1, . . . , k}, where the sets of colors assigned to any two vertices distance d apart share fewer than d colors in common. The minimum integer k such that a graph G has a t- tone k-coloring is known as the t-tone chromatic number. We study the 2-tone chromatic number in three different graph products. In particular, given graphs G and H, we bound the 2-tone chromatic number for the direct product G×H, the Cartesian product G□H, and the strong product G⊠H.

LA - eng

KW - t-tone coloring; Cartesian product; direct product; strong product; -tone coloring; cartesian product

UR - http://eudml.org/doc/271235

ER -

## References

top- [1] A. Bickle and B. Phillips, t-tone colorings of graphs, submitted (2011).
- [2] D. Cranston, J. Kim and W. Kinnersley, New results in t-tone colorings of graphs, Electron. J. Comb. 20(2) (2013) #17. Zbl1266.05009
- [3] D. Bal, P. Bennett, A. Dudek and A. Frieze, The t-tone chromatic number of random graphs, Graphs Combin. 30 (2013) 1073-1086. doi:10.1007/s00373-013-1341-9[Crossref][WoS] Zbl1298.05100
- [4] N. Fonger, J. Goss, B. Phillips and C. Segroves, Math 6450: Final Report, (2011). http://homepages.wmich.edu/~zhang/finalReport2.pdf
- [5] D. West, REGS in Combinatorics. t-tone colorings, (2011). http://www.math.uiuc.edu/~west/regs/ttone.html
- [6] R. Hammack, W. Imrich and S. Klavˇzar, Handbook of Product Graphs, Second Edition (CRC Press, Boca Raton, 2011). Zbl1283.05001
- [7] S. Krumke, M. Marathe and S. Ravi, Approximation algorithms for channel assignment in radio networks, Wireless Networks 7 (2001) 575-584. doi:10.1023/A:1012311216333[Crossref] Zbl0996.68009
- [8] V. Vazirani, Approximation Algorithms (Springer, 2001). Zbl0999.68546

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