Abstract
The channel assignment is an important problem with applications in optical networks. This problem was formulated to the L(p, 1)-labeling of graphs by Griggs and Yeh. The r-dynamic coloring is a generalization of the L(1, 1)-labeling. An r-dynamic k-coloring of a graph G is a proper k-coloring such that every vertex v is adjacent to at least min\(\{d(v),r\}\) different colors. Denote \(\chi _{r}(G)=min\{k \mid G\) has an r-dynamic k-coloring} and \(ch_{r}(G)=min\{k \mid G\) has a list r-dynamic k-coloring}. In this paper, we show upper bounds \(ch_{r}(G)\le r+5\) for planar graphs G with \(g(G)\ge 5\) and \(r\ge 15\), \(ch_{r}(G)\le r+10\) for graphs G with \(mad(G)<\frac{10}{3}\).
Keywords
Supported by NSFC 11771403.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Bondy, J.A., Murty, U.S.R.: Graph Theory. Springer, New York (2008)
Bu, Y.H., Zhu, X.B.: An optimal square coloring of planar graphs. J. Comb. Optim. 24, 580–592 (2012)
Cranston, D.W., Kim, S.J.: List-coloring the square of a subcubic graph. J. Graph Theory 57, 65–87 (2008)
Chen, Y., Fan, S.H., Lai, H.J., Song, H.M., Sun, L.: On dynamic coloring for planar graphs and graphs of higher genus. Discrete Appl. Math. 160, 1064–1071 (2012)
Ding, C., Fan, S.H., Lai, H.J.: Upper bound on conditional chromatic number of graphs. Jinan Univ. 29, 7–14 (2008)
Griggs, J.R., Yeh, R.K.: Labelling graphs with a condition at distance 2. SIAM J. Discrete Math. 5, 586–595 (1992)
Jahanbekam, S., Kim, J., Suil, O., West, D.B.: On r-dynamic colorings of graphs. Discrete Appl. Math. 206, 65–72 (2016)
Kim, S.J., Lee, S.J., Park, W.J.: Dynamic coloring and list dynamic coloring of planar graphs. Discrete Appl. Math. 161, 2207–2212 (2013)
Kim, S.-J., Park, W.-J.: List dynamic coloring of sparse graphs. In: Wang, W., Zhu, X., Du, D.-Z. (eds.) COCOA 2011. LNCS, vol. 6831, pp. 156–162. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-22616-8_13
Kim, S.J., Park, B.: List 3-dynamic coloring of graphs with small maximum average degree. Discrete Math. 341, 1406–1418 (2018)
Lai, H.J., Lin, J., Montgomery, B., Tao, Z., Fan, S.H.: Conditional colorings of graphs. Discrete Math. 306, 1997–2004 (2006)
Lai, H.J., Montgomery, B., Poon, H.: Upper bounds of dynamic coloring number. Ars Combin. 68, 193–201 (2003)
Li, H., Lai, H.J.: 3-dynamic coloring and list 3-dynamic coloring of K1,3-free graphs. Discrete Appl. Math. 222, 166–171 (2017)
Lin,Y.: Upper bounds of conditional chromatics number. Master Thesis, Jinan University (2008)
Loeb, S., Mahoney, T., Reiniger, B., Wise, J.: Dynamic coloring parameters for grphs with given genus. Discrete Appl. Math. 235, 129–141 (2018)
Montgomery, B.: (PhD Dissertation). West Virginia University (2001)
Roberts, F.S.: T-colorings of graphs: recent results and open problems. Discrete Math. 93, 229–245 (1991)
Song, H.M., Fan, S.H., Chen, Y., Sun, L., Lai, H.J.: On r-hued coloring of K4-minor free graphs. Discrete Math. 315–316, 47–52 (2014)
Song, H.M., Lai, H.J., Wu, J.L.: On r-hued coloring of planar graphs with girth at least 6. Discrete Appl. Math. 198, 251–263 (2016)
Wegner, G.: Graphs with given diameter and a coloring problem. Technical report. University of Dortmund (1997)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer Nature Switzerland AG
About this paper
Cite this paper
Zhu, J., Bu, Y. (2018). Channel Assignment with r-Dynamic Coloring. In: Tang, S., Du, DZ., Woodruff, D., Butenko, S. (eds) Algorithmic Aspects in Information and Management. AAIM 2018. Lecture Notes in Computer Science(), vol 11343. Springer, Cham. https://doi.org/10.1007/978-3-030-04618-7_4
Download citation
DOI: https://doi.org/10.1007/978-3-030-04618-7_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-04617-0
Online ISBN: 978-3-030-04618-7
eBook Packages: Computer ScienceComputer Science (R0)