### The locating-chromatic number and partition dimension of fibonacene graphs

#### Abstract

Fibonacenes are unbranched catacondensed benzenoid hydrocarbons in which all the non-terminal hexagons are angularly annelated. A hexagon is said to be angularly annelated if the hexagon is adjacent to exactly two other hexagons and possesses two adjacent vertices of degree 2. Fibonacenes possess remarkable properties related with Fibonacci numbers. Various graph properties of fibonacenes have been extensively studied, such as their saturation numbers, independence numbers and Wiener indices. In this paper, we show that the locating-chromatic number of any fibonacene graph is 4 and the partition dimension of such a graph is 3.

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P.G. Anderson, Fibonaccene, in: A. N. Philippou, G. E. Bergum, and A. F. Horadam (Eds.), Fibonacci numbers and their applications, Reidel, Dordrecht, (1986), pp. 1—8.

Asmiati and E.T. Baskoro, Characterizing all graphs containing cycle with the locating-chromatic number 3, AIP Conf. Proc., 1450 (2012), 351-–357.

Asmiati, Wamiliana, Devriyadi and R. Yulianti, On some Petersen graphs having locating-chromatic number four or five, FJMS, 102 (2017), No.4, 769–778.

A.T. Balaban, Chemical graphs. 50. Symmetry and enumeration of fibonacenes (Unbranched catacondensed benzenoids isoarithmic with helicenes and zigzag catafusenes) MATCH Com-mun. Math. Comput. Chem., 24 (1989), 29–38.

E.T. Baskoro and Asmiati, Characterizing all trees with locating-chromatic number 3, EJGTA, 1(2) (2013), 109–117.

G. Chartrand, L. Eroh, M. A. Johnson, and O. R. Oellermann, Resolvability in graphs and the metric dimension of a graph, Discrete Appl. Math., 105 (2000), 99–113.

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.

G. Chartrand, E. Salehi and P. Zhang, On the partition dimension of a graph, Congr. Numer., 130 (1998), 157–168.

G. Chartrand, E. Salehi and P. Zhang, The partition dimension of a graph, Aequationes Math., 59 (2000), 45–54.

A. A. Dobrynin, I. Gutman, S. Klavˇzar and P.ˇZigert, Wiener index of hexagonal systems, Acta Applicandae Mathematicae, 72 (2002), 247—294.

T. Doˇsli´ c and I. Zubac, Saturation number of benzenoid graphs, MATCH Commun. Math. Comput. Chem., 73 (2015), 491–500.

S. Fajtlowicz, Pony Express—Graffiti’s conjectures about carcinogenic and stable ben-zenoids.

M. Ghanem, H. Al-Ezah and A. Dabbour, Locating chromatic number of powers of paths and cycles, symmetry, 11 (2019), 389–396.

I. Gutman and S. Klavˇ zar, Chemical graph theory of fibonacenes, MATCH Commun. Math.Comput. Chem., 55 (2006), 39–54.

F. Harary and R. A. Melter, On the metric dimension of a graph, Ars Combin, 2 (1976), 191–195.

R. Pepper, An upper bound on the independence number of benzenoid systems, Discrete App.Math, 156 (2008), 607–619.

I.A. Purwasih and E.T. Baskoro, The locating-chromatic number for Halin graphs, AIP Conf.Proc., 1450 (2012), 342–345.

I.A. Purwasih, E.T. Baskoro, H. Assiyatun and W. Djohan, The locating-chromatic number for a subdivision of a wheel on one cycle edge, AKCE Int. J. Graphs Comb., 10 (2013),No.3, 327–336.

I. Tomescu, Discrepancies between metric dimension and partition dimension of a connected graph, Discrete Math., 308 (2008), 5026–5031.

N. Trinajsti´ c, Chemical Graph Theory: Second Edition, CRC Press, Tokyo/London, 1983.

DOI: http://dx.doi.org/10.19184/ijc.2019.3.2.5

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