A Maximum Resonant Set of Polyomino Graphs

Open access

Abstract

A polyomino graph P is a connected finite subgraph of the infinite plane grid such that each finite face is surrounded by a regular square of side length one and each edge belongs to at least one square. A dimer covering of P corresponds to a perfect matching. Different dimer coverings can interact via an alternating cycle (or square) with respect to them. A set of disjoint squares of P is a resonant set if P has a perfect matching M so that each one of those squares is M-alternating. In this paper, we show that if K is a maximum resonant set of P, then P − K has a unique perfect matching. We further prove that the maximum forcing number of a polyomino graph is equal to the cardinality of a maximum resonant set. This confirms a conjecture of Xu et al. [26]. We also show that if K is a maximal alternating set of P, then P − K has a unique perfect matching.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1] H. Abeledo and G.W. Atkinson Unimodularity of the Clar number problem Linear Algebra Appl. 420 (2007) 441-448. doi:10.1016/j.laa.2006.07.026

  • [2] C. Berge C.C. Chen V. Chvátal and C.S. Seow Combinatorial properties of polyominoes Combinatorica 1 (1981) 217-224. doi:10.1007/BF02579327

  • [3] Z. Che and Z. Chen Forcing on perfect matchings - A survey MATCH Commun. Math. Comput. Chem. 66 (2011) 93-136.

  • [4] E. Clar The Aromatic Sextet (Wiley London 1972).

  • [5] M.E. Fisher Statistical mechanics of dimers on a plane lattice Phys. Rev. 124 (1961) 1664-1672. doi:10.1103/PhysRev.124.1664

  • [6] E.J. Cockayne Chessboard domination problems Discrete Math. 86 (1990) 13-20. doi:10.1016/0012-365X(90)90344-H

  • [7] C.M. Grinstead B. Hahne and D. Van Stone On the queen domination problem Discrete Math. 86 (1990) 21-26. doi:10.1016/0012-365X(90)90345-I

  • [8] I. Gutman S.J. Cyvin Advances in the Theory of Benzenoid Hydrocarbons (Springer Berlin 1990). doi:10.1007/3-540-51505-4

  • [9] F. Harary D.J. Klein and T.P. Živkovič Graphical properties of polyhexes: Perfect matching vector and forcing J. Math. Chem. 6 (1991) 295-306. doi:10.1007/BF01192587

  • [10] W.C. Herndon Resonance energies of aromatic hydrocarbons: Quantitative test of resonance theory J. Am. Chem. Soc. 95 (1973) 2404-2406. doi:10.1021/ja00788a073

  • [11] P.W. Kasteleyn The statistics of dimers on a lattice: I. The number of dimer arrangements on a quadratic lattice Physica 27 (1961) 1209-1225. doi:10.1016/0031-8914(61)90063-5

  • [12] X. Ke A lower bound on the number of elementary components of essentially disconnected generalized polyomino graphs J. Math. Chem. 50 (2012) 131-140. doi:10.1007/s10910-011-9900-x

  • [13] D.J. Klein and M. Randić Innate degree of freedom of a graph J. Comput. Chem. 8 (1987) 516-521. doi:10.1002/jcc.540080432

  • [14] W. Li and H. Zhang Dimer statistics of honeycomb lattices on Klein bottle Möbius strip and cylinder Phys. A 391 (2012) 3833-3848. doi:10.1016/j.physa.2012.03.004

  • [15] S. Liu and J. Ou On maximal resonance of polyomino graphs J. Math. Chem. 51 (2013) 603-619. doi:10.1007/s10910-012-0104-9

  • [16] F. Lu and L. Zhang Dimers on two types of lattices on the Klein bottle J. Phys. A 45 (2012) #49. doi:10.1088/1751-8113/45/49/494012

  • [17] L. Lovász and M.D. Plummer Matching Theory (Annals of Discrete Mathematics Vol. 29 North-Holland Amsterdam 1986).

  • [18] A. Motoyama and H. Hosoya King and domino polynomials for polyomino graphs J. Math. Phys. 18 (1977) 1485-1490. doi:10.1063/1.523411

  • [19] L. Pachter and P. Kim Forcing matchings on square grids Discrete Math. 190 (1998) 287-294. doi:10.1016/S0012-365X(97)00266-5

  • [20] M. Randić Conjugated circuits and resonance energies of benzenoid hydrocarbons Chem. Phys. Lett. 38 (1976) 68-70. doi:10.1016/0009-2614(76)80257-6

  • [21] M. Randić Aromaticity and conjugation J. Am. Chem. Soc. 99 (1977) 444-450. doi:10.1021/ja00444a022

  • [22] H. Sachs Perfect matchings in hexagonal systems Combinatorica 4 (1980) 89-99. doi:10.1007/BF02579161

  • [23] H. Sachs and H. Zernitz Remark on the dimer problem Discrete Appl. Math. 51 (1994) 171-179. doi:10.1016/0166-218X(94)90106-6

  • [24] K. Salem and H. Abeledo A maximal alternating set of a hexagonal system MATCH Commun. Math. Comput. Chem. 55 (2006) 159-176.

  • [25] S. Wei and X. Ke Elementary components of essentially disconnected polyomino graphs J. Math. Chem. 47 (2010) 496-504. doi:10.1007/s10910-009-9589-2

  • [26] L. Xu H. Bian and F. Zhang Maximum forcing number of hexagonal systems MATCH Commun. Math. Comput. Chem. 70 (2013) 493-500.

  • [27] W. Yan Y.-N. Yeh and F. Zhang Dimer problem on the cylinder and torus Phys. A 387 (2008) 6069-6078. doi:10.1016/j.physa.2008.06.042

  • [28] F. Zhang X. Guo and R. Chen The connectivity of Z-transformation graphs of perfect matchings of hexagonal systems Acta Math. Appl. Sin. 4 (1988) 131-135. doi:10.1007/bf02006061

  • [29] H. Zhang The connectivity of Z-transformation graphs of perfect matchings of polyominoes Discrete Math. 158 (1996) 257-272. doi:10.1016/0012-365X(95)00048-2

  • [30] H. Zhang and F. Zhang Perfect matchings of polyomino graphs Graphs Combin. 13 (1997) 295-304. doi:10.1007/BF03353008

  • [31] H. Zhang and F. Zhang Plane elementary bipartite graphs Discrete Appl. Math. 105 (2000) 291-311. doi:10.1016/S0166-218X(00)00204-3

  • [32] M. Zheng and R. Chen A maximal cover of hexagonal systems Graphs Combin. 1 (1985) 295-298. doi:10.1007/BF02582955

Search
Journal information
Impact Factor
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 audience:

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

Metrics
All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 198 152 7
PDF Downloads 70 50 2