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continuous; $F$-continuous; $F$-regular; regular graph
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.
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