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Title: Results on $F$-continuous graphs (English)
Author: Draganova, Anna
Language: English
Journal: Czechoslovak Mathematical Journal
ISSN: 0011-4642 (print)
ISSN: 1572-9141 (online)
Volume: 59
Issue: 1
Year: 2009
Pages: 51-60
Summary lang: English
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Category: math
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Summary: For any nontrivial connected graph $F$ and any graph $G$, the {\it $F$-degree} of a vertex $v$ in $G$ is the number of copies of $F$ in $G$ containing $v$. $G$ is called {\it $F$-continuous} if and only if the $F$-degrees of any two adjacent vertices in $G$ differ by at most 1; $G$ is {\it $F$-regular} if the $F$-degrees of all vertices in $G$ are the same. This paper classifies all $P_4$-continuous graphs with girth greater than 3. We show that for any nontrivial connected graph $F$ other than the star $K_{1,k}$, $k \geq 1$, there exists a regular graph that is not $F$-continuous. If $F$ is 2-connected, then there exists a regular $F$-continuous graph that is not $F$-regular. (English)
Keyword: continuous
Keyword: $F$-continuous
Keyword: $F$-regular
Keyword: regular graph
MSC: 05C12
MSC: 05C78
idZBL: Zbl 1224.05434
idMR: MR2486615
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Date available: 2010-07-20T14:50:52Z
Last updated: 2016-04-07
Stable URL: http://hdl.handle.net/10338.dmlcz/140463
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Reference: [1] Chartrand, G., Eroh, L., Schultz, M., Zhang, P.: An introduction to analytic graph theory.Util. Math. 59 (2001), 31-55. Zbl 0989.05035, MR 1832600
Reference: [2] Chartrand, G., Holbert, K. S., Oellermann, O. R., Swart, H. C.: $F$-Degrees in graphs.Ars Comb. 24 (1987), 133-148. Zbl 0643.05055, MR 0917968
Reference: [3] Chartrand, G., Jarrett, E., Saba, F., Salehi, E., Zhang, P.: $F$-Continuous graphs.Czech. Math. J. 51 (2001), 351-361. Zbl 0977.05042, MR 1844315, 10.1023/A:1013751031651
Reference: [4] Erdös, P., Sachs, H.: Reguläre Graphen gegebener Taillenweite mit minimaler Knotenzahl.Wiss Z. Martin-Luther-Univ. Halle-Wittenberg, Math.-Naturwiss. Reihe 12 (1963), 251-258 . MR 0165515
Reference: [5] Gimbel, J., Zhang, P.: Degree-continuous graphs.Czech. Math. J. 51 (2001), 163-171. Zbl 1079.05504, MR 1814641
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