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Keywords:
continuous; $F$-continuous; $F$-regular; regular graph
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.
References:
[1] Chartrand, G., Eroh, L., Schultz, M., Zhang, P.: An introduction to analytic graph theory. Util. Math. 59 (2001), 31-55. MR 1832600 | Zbl 0989.05035
[2] Chartrand, G., Holbert, K. S., Oellermann, O. R., Swart, H. C.: $F$-Degrees in graphs. Ars Comb. 24 (1987), 133-148. MR 0917968 | Zbl 0643.05055
[3] Chartrand, G., Jarrett, E., Saba, F., Salehi, E., Zhang, P.: $F$-Continuous graphs. Czech. Math. J. 51 (2001), 351-361. DOI 10.1023/A:1013751031651 | MR 1844315 | Zbl 0977.05042
[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
[5] Gimbel, J., Zhang, P.: Degree-continuous graphs. Czech. Math. J. 51 (2001), 163-171. MR 1814641 | Zbl 1079.05504

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