Abstract
For any nontrivial connected graph F and any graph G, the F-degree of a vertex v in G is the number of copies of F in G containing v. G is called F-continuous if and only if the F-degrees of any two adjacent vertices in G differ by at most 1; G is 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 ⩾ 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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Draganova, A. Results on F-continuous graphs. Czech Math J 59, 51–60 (2009). https://doi.org/10.1007/s10587-009-0004-8
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DOI: https://doi.org/10.1007/s10587-009-0004-8