For a nontrivial connected graph $F$, the $F$-degree of a vertex $v$ in a graph $G$ is the number of copies of $F$ in $G$ containing $v$. A graph $G$ is $F$-continuous (or $F$-degree continuous) if the $F$-degrees of every two adjacent vertices of $G$ differ by at most 1. All $P_3$-continuous graphs are determined. It is observed that if $G$ is a nontrivial connected graph that is $F$-continuous for all nontrivial connected graphs $F$, then either $G$ is regular or $G$ is a path. In the case of a 2-connected graph $F$, however, there always exists a regular graph that is not $F$-continuous. It is also shown that for every graph $H$ and every 2-connected graph $F$, there exists an $F$-continuous graph $G$ containing $H$ as an induced subgraph.
A vertex coloring of a graph G is a multiset coloring if the multisets of colors of the neighbors of every two adjacent vertices are different. The minimum k for which G has a multiset k-coloring is the multiset chromatic number χm(G) of G. For every graph G, χm(G) is bounded above by its chromatic number χ(G). The multiset chromatic number is determined for every complete multipartite graph as well as for cycles and their squares, cubes, and fourth powers. It is conjectured that for each k ≥ 3, there exist sufficiently large integers n such that χm(C k n) = 3. It is determined for which pairs k, n of integers with 1 ≤ k ≤ n and n ≥ 3, there exists a connected graph G of order n with χm(G) = k. For k = n − 2, all such graphs G are determined.