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\(F\)-degrees in graphs. (English) Zbl 0643.05055

Let \(F\) and \(G\) be graphs. The \(F\)-degree of a vertex \(v\) in \(G\), denoted by \(F\)-\(\deg v\) is the number of subgraphs of \(G\) isomorphic to \(F\) that contain \(v\). For example, the \(K_ 2\)-degree of a vertex \(v\) in any graph \(G\) is simply the degree of \(v\) in \(G\). A graph \(G\) is \(F\)-regular of degree \(k\) if every vertex of \(G\) has \(F\)-degree \(k\). The four authors investigate \(F\)-regular graphs and specify some classes of \(F\)-regular graphs. Furthermore, they prove the existence of \(F\)-irregular graphs for special \(F\) and conjecture that there exists an \(F\)-irregular graph for every connected graph \(F\) of order at least 3.
Reviewer: R.Bodendiek

MSC:

05C99 Graph theory
05C75 Structural characterization of families of graphs

Keywords:

regular graphs
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