硼氮富勒烯圖環(huán)4-邊割的刻畫

蔣曉艷

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硼氮富勒烯圖環(huán)4-邊割的刻畫

蔣曉艷

（惠州學(xué)院 數(shù)學(xué)系，廣東 惠州 516007）

硼氮富勒烯圖是連通、正則的平面圖，且每個(gè)面要么是四邊形，要么是六邊形. 本文刻畫了有非平凡環(huán)邊割的硼氮富勒烯圖，即有非平凡環(huán)邊割，則是一類管狀圖，或的一個(gè)分支是2個(gè)相鄰的四邊形，或的一個(gè)分支是3個(gè)相鄰的四邊形（即第1個(gè)與第2個(gè)相鄰，第2個(gè)與第3個(gè)相鄰，但第1個(gè)與第3個(gè)不相鄰）.

硼氮富勒烯圖；環(huán)邊連通度；環(huán)邊割

1 引言及預(yù)備知識(shí)

圖1 示意圖

2 有非平凡環(huán)4-邊割的硼氮富勒烯圖的刻畫

文獻(xiàn)[10]計(jì)算了硼氮富勒烯圖的環(huán)邊連通度，結(jié)果如下.

定理1[10]109對(duì)硼氮富勒烯圖，若，則的環(huán)邊連通度為3，否則為4.

由定理1知，硼氮富勒烯圖環(huán)邊連通度為3的圖只有一類管狀圖，所以下面只需討論環(huán)邊連通度為4的圖的非平凡環(huán)邊割.

a.示意圖 b.帽子 c.示意圖

關(guān)于硼氮富勒烯圖的面圈有以下結(jié)論：

引理1[11]1892設(shè)是硼氮富勒烯圖的一個(gè)長(zhǎng)圈，則是一個(gè)面圈的邊界.

引理2[11]1892設(shè)為硼氮富勒烯圖. 若是的一個(gè)長(zhǎng)圈，則是

1）一個(gè)六角形面的邊界，或

2）2個(gè)相鄰四邊形面的邊界，或

3）有一個(gè)公共點(diǎn)的3個(gè)相鄰四邊形面的邊界.

下面給出本文的主要結(jié)論.

圖3 示意圖

另一方面，根據(jù)點(diǎn)、邊、面的關(guān)系可以得到：

a.有4個(gè)相繼度點(diǎn) b.有3個(gè)相繼度點(diǎn) c.有3個(gè)相繼度點(diǎn)

a.有2個(gè)相鄰度點(diǎn),和,b.有2個(gè)相鄰度點(diǎn),和,

c.有2個(gè)相鄰度點(diǎn), d.有2個(gè)相鄰度點(diǎn),

圖6 子情況3.4示意圖

綜合以上分析，我們可得到定理2的結(jié)論.

[1] FOWLER P, HELNE T, MITCHELL D, et al. Boron-nitrogen analogues of fullerenes: the isolated-square rule [J]. J Chem Soc Faraday Trans, 1996, 92(12): 2197-2201.

[2] LIN Chengde, TANG Peng. Kekule count in capped zigzag B-N nanotubes [J]. J Chem Inf Comput Sci, 2004, 44: 13-20.

[3] SHIU W C, LAM P C B, ZHANG Heping. Clar and sextet polynomials of buckminster-Fullerene [J]. J Mol Struct (Theochem), 2003, 622: 239-248.

[4] STROUT D. Fullerene-like cages versus alternant cages: isomer stability of B13N13, B14N14 and B16N16 [J]. Chem Phys Lett, 2004, 383: 95-98.

[5] ZHU Hongyao, KLEIN D, SEITZ W, et al. B-N alternants: boron nitride cages and polymers [J]. Inorg Chem, 1995, 34: 1377-1383.

[6] KOCHOL M. A cyclically 6-edge-connected snark of order 118 [J]. Discrete Math, 1996, 161: 297-300.

[7] LOU Dingjun, HOLTON D A. Lower bound of cyclic edge connectivity for-extendability of regular graphs [J]. Discrete Math, 1993, 112: 139-150.

[8] NEDELA R, SKOVIERA M. Atoms of cyclic connectivity in cubic graphs [J]. Math Slovaca, 1995, 45: 481-499.

[9] KUTNAR K, MARUSIC D. On cyclic edge-connectivity of fullerenes [J]. Discrete Appl Math, 2008, 156: 1661-1669.

[10] DOSLIC T. Cyclical edge-connectivity of fullerene graphs and (,6)-cages [J]. J Math Chem, 2003, 33: 103-111.

[11] JIANG Xiaoyan, ZHANG Heping. On forcing matching number of boron-nitrogen fullerene graphs [J]. Discrete Appl Math, 2001, 159: 1581-1593.

[責(zé)任編輯：熊玉濤]

Characterization of Cyclic 4-edge Cut of Boron-nitrogen Fullerene Graphs

JIANGXiao-yan

(Department of Mathematics, Huizhou University, Huizhou 516007, China)

A boron-nitrogen fullerene graph is a 3-connected, 3-regular planar graph, and each face is square or hexagon. In this paper, we characterize boron-nitrogen fullerene graphwith non-trivial cyclical-4-edge cut, that is,has non-trivial cyclical-4-edge cut, thenis a type of tubulous graph, or a component ofis two adjacent squares, or a component ofis three adjacent squares (the first is adjacent to the second square, the second is adjacent the third square, but the first is not adjacent to the third square.)

boron-nitrogen fullerene graph; cyclical-edge connectivity; cyclical-edge cut

1006-7302（2016）02-0009-05

O157.5

A

2015-12-24

廣東省普通高校青年創(chuàng)新人才項(xiàng)目（2015KQNCX152）；惠州學(xué)院博士啟動(dòng)基金（C5110208）

蔣曉艷（1982—），女，山東青島人，講師，博士，研究方向?yàn)閼?yīng)用數(shù)學(xué).