### Locating-chromatic number of the edge-amalgamation of trees

#### Abstract

The investigation on the locating-chromatic number of a graph was initiated by Char- trand et al. (2002). This concept is in fact a special case of the partition dimension of a graph. This topic has received much attention. However, the results are still far from satisfaction. We can define the locating-chromatic number of a graph G as the smallest integer k such that there exists a k-partition of the vertex-set of G such that all vertices have distinct coordinates with respect to this partition. As we know that the metric dimension of a tree is completely solved. However, the locating-chromatic numbers for most of trees are still open. For i = 1, 2, . . . , t, let Ti be a tree with a fixed edge eoi called the terminal edge. The edge-amalgamation of all Tis denoted by Edge-Amal{Ti;eoi} is a tree formed by taking all the Tis and identifying their terminal edges. In this paper, we study the locating-chromatic number of the edge-amalgamation of arbitrary trees. We give lower and upper bounds for their locating-chromatic numbers and show that the bounds are tight.

#### Keywords

#### Full Text:

PDFDOI: http://dx.doi.org/10.19184/ijc.2020.4.2.6

#### References

Asmiati, H. Assiyatun, and E.T. Baskoro, Locating-chromatic number of amalgamation of stars, *ITB J. Sci.* 43 A (1) (2011), 1–8.

Asmiati, E.T. Baskoro, H. Assiyatun, D. Suprijanto, R. Simanjuntak, and S. Uttunggadewa, The locating-chromatic number of firecracker graphs, *Far East J. Math. Sci.* 63:1 (2012), 11–23.

Asmiati, Bilangan kromatik lokasi graf pohon dan karakterisasi graf dengan bilangan kromatik lokasi 3, *Disertasi Program Studi Doktor Matematika, Institut Teknologi Bandung*, 2012 (in Indonesian).

E.T. Baskoro and Asmiati, Characterizing all trees with locating-chromatic number 3, *Electron. J. Graph Theory Appl.* 1 (2) (2013), 109–117.

G. Chartrand, D. Erwin, M.A. Henning, P.J. Slater, and P. Zhang, The locating-chromatic number of a graph, *Bull. Inst. Combin. Appl.* 36 (2002), 89–101.

D.K. Syofyan, E.T. Baskoro, and H. Assiyatun, On the locating-chromatic number of homogeneous lobster, *AKCE Int. J. Graphs Comb.* 10 (3) (2013), 245–252.

D.K. Syofyan, E.T. Baskoro, and H. Assiyatun, The locating-chromatic number of binary trees, *Elsevier, The 2nd International Conference of Graph Theory and Information Security*, 74 (2015), 79–83.

D.K. Syofyan, E.T. Baskoro, and H. Assiyatun, Trees with certain locating-chromatic number, *J. Math. Fund. Sci.* 48 (1) (2016), 39–47.

D. Welyyanti, E.T. Baskoro, R. Simanjuntak, and S. Uttunggadewa, On locating-chromatic number of complete n-ary tree, *AKCE Int. J. Graphs Combin*. 3 (2013), 309–315.

### Refbacks

- There are currently no refbacks.

This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.