Title:
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An upper bound on the basis number of the powers of the complete graphs (English) |
Author:
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Alsardary, Salar Y. |
Language:
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English |
Journal:
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Czechoslovak Mathematical Journal |
ISSN:
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0011-4642 (print) |
ISSN:
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1572-9141 (online) |
Volume:
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51 |
Issue:
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2 |
Year:
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2001 |
Pages:
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231-238 |
Summary lang:
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English |
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Category:
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math |
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Summary:
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The basis number of a graph $G$ is defined by Schmeichel to be the least integer $h$ such that $G$ has an $h$-fold basis for its cycle space. MacLane showed that a graph is planar if and only if its basis number is $\le 2$. Schmeichel proved that the basis number of the complete graph $K_n$ is at most $3$. We generalize the result of Schmeichel by showing that the basis number of the $d$-th power of $K_n$ is at most $2d+1$. (English) |
MSC:
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05C10 |
MSC:
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05C35 |
MSC:
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05C38 |
MSC:
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05C99 |
idZBL:
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Zbl 0977.05134 |
idMR:
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MR1844307 |
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Date available:
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2009-09-24T10:41:57Z |
Last updated:
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2020-07-03 |
Stable URL:
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http://hdl.handle.net/10338.dmlcz/127644 |
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Reference:
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[1] A. A. Ali and Salar Y. Alsardary: On the basis number of a graph.Dirasat 14 (1987), 43–51. |
Reference:
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[2] A. A. Ali: The basis number of the join of graphs.Arab J. Math. 10 (1989), 21–33. Zbl 0806.05043 |
Reference:
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[3] A. A. Ali: The basis number of complete multipartite graphs.Ars Combin. 28 (1989), 41–49. Zbl 0728.05058, MR 1039129 |
Reference:
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[4] A. A. Ali: The basis number of the direct products of paths and cycles.Ars Combin. 27 (1989), 155–164. MR 0989461 |
Reference:
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[5] A. A. Ali and G. T. Marougi: The basis number of the cartesian product of some graphs.J. Indian Math. Soc. (N.S.) 58 (1992), 123–134. MR 1207033 |
Reference:
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[6] Salar Y. Alsardary and A. A. Ali: The basis number of some special non-planar graphs.Czechoslovak Math. J (to appear). MR 1983447 |
Reference:
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[7] J. A. Banks and E. F. Schmeichel: The basis number of the $n$-cube.J. Combin. Theory, Ser. B 33 (1982), 95–100. MR 0685059, 10.1016/0095-8956(82)90061-2 |
Reference:
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[8] J. A. Bondy and S. R. Murty: Graph Theory with Applications.Amer. Elsevier, New York, 1976. MR 0411988 |
Reference:
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[9] S. MacLane: A combinatorial condition for planar graphs.Fund. Math. 28 (1937), 22–32. Zbl 0015.37501 |
Reference:
|
[10] E. F. Schmeichel: The basis number of a graph.J. Combin. Theory, Ser. B. 30 (1981), 123–129. Zbl 0385.05031, MR 0615307, 10.1016/0095-8956(81)90057-5 |
Reference:
|
[11] J. Wojciechowski: Long snakes in powers of the complete graph with an odd number of vertices.J. London Math. Soc. II, Ser. 50 (1994), 465–476. Zbl 0814.05043, MR 1299451, 10.1112/jlms/50.3.465 |
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