# Article

 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 . Category: math . 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 . Date available: 2010-07-20T14:50:52Z Last updated: 2016-04-07 Stable URL: http://hdl.handle.net/10338.dmlcz/140463 . 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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