Sharp Upper Bounds on the Clar Number of Fullerene Graphs

Open access


The Clar number of a fullerene graph with n vertices is bounded above by ⌊n/6⌋ − 2 and this bound has been improved to ⌊n/6⌋ − 3 when n is congruent to 2 modulo 6. We can construct at least one fullerene graph attaining the upper bounds for every even number of vertices n ≥ 20 except n = 22 and n = 30.

[1] P.W. Fowler and D.E. Manolopoulos, An Atlas of Fullerenes (Oxford University Press, Oxford, 1995).

[2] Y. Gao and H. Zhang, The Clar number of fullerenes on surfaces, MATCH Commun. Math. Comput. Chem. 72 (2014) 411–426.

[3] Y. Gao and H. Zhang, Clar structure and Fries set of fullerenes and (4, 6)-fullerenes on surfaces, J. Appl. Math. (2014) Article ID 196792, 11 pages.

[4] Y. Gao, Q. Li and H. Zhang, Fullerenes with the maximum Clar number, Discrete Appl. Math. 202 (2016) 58–69. doi:10.1016/j.dam.2015.08.007

[5] C. Godsil and G. Royle, Algebraic Graph Theory (Springer-Verlag, New York, 2001). doi:10.1007/978-1-4613-0163-9

[6] B. Grünbaum and T. Motzkin, The number of hexagons and the simplicity of geodesics on certain polyhedra, Canad. J. Math. 15 (1963) 744–751. doi:10.4153/CJM-1963-071-3

[7] E. Hartung, Fullerenes with complete Clar structure, Discrete Appl. Math. 161 (2013) 2952–2957. doi:10.1016/j.dam.2013.06.009

[8] T. Pisanski and M. Randić, Bridges between geometry and graph theory, in: Geometry at Work: Papers in Applied Geometry Vol. 53, C.A. Gorini (Ed(s)), (Washington, DC, Mathematical Association of America, 2000) 174–194.

[9] D. Ye and H. Zhang, Extremal fullerene graphs with the maximum Clar number, Discrete Appl. Math. 157 (2009) 3152–3173. doi:10.1016/j.dam.2009.06.007

[10] H. Zhang and F. Zhang, The Clar covering polynomial of hexagonal systems I, Discrete Appl. Math. 69 (1996) 147–167. doi:10.1016/0166-218X(95)00081-2

[11] H. Zhang and D. Ye, An upper bound for Clar number of fullerene graphs, J. Math. Chem. 42 (2007) 123–133. doi:10.1007/s10910-006-9061-5

[12] H. Zhang, D. Ye and Y. Liu, A combination of Clar number and Kekulé count as an indicator of relative stability of fullerene isomers of C60, J. Math. Chem. 48 (2010) 733–740. doi:10.1007/s10910-010-9706-2

Discussiones Mathematicae Graph Theory

The Journal of University of Zielona Góra

Journal Information

IMPACT FACTOR 2018: 0.741
5-year IMPACT FACTOR: 0.611

CiteScore 2018: 0.73

SCImago Journal Rank (SJR) 2018: 0.763
Source Normalized Impact per Paper (SNIP) 2018: 0.934

Mathematical Citation Quotient (MCQ) 2017: 0.36

Target Group

researchers in the fields of: colourings, partitions (general colourings), hereditary properties, independence and dominating structures (sets, paths, cycles, etc.), cycles, local properties, products of graphs


All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 233 191 8
PDF Downloads 91 75 4