F-Continuous Graphs

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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 P3-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.

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