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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 657.05048

**Autor: ** Erdös, Paul; Faudree, Ralph J.; Gyárfás, A.; Schelp, R.H.

**Title: ** Cycles in graphs without proper subgraphs of minimum degree 3. (In English)

**Source: ** Ars Comb. 25B, 195-201 (1988).

**Review: ** Let G(n,m) denote the set of graphs with n vertices and m edges. It is well-kown that each G in G(n,2n-2) contains a subgraph of minimum degree 3 but there is a G in G(n,2n-3) with no subgraph of minimum degree 3. Furthermore, each G in G(n,2n-1) contains a proper subgraph of minimum degree3 but there is a G in G(n,2n-2) without this property. Here the authors primarily study cycle lengths of graphs in G^*(n,2n-2), where G^*(n,2n-2) is the set of graphs with n vertices, 2n-2 edges, and no proper subgraph with minimum degree 3. it is shown, for example, that if G in G^*(n,2n-2) and n \geq 6, then G contains a C_{m} for m = 3,4,5. Such graphs also contain ``long'' cycles. It is shown that if G in G^*(n,2n-2) then G contains a cycle of length at least \lfloor log n\rfloor.

**Reviewer: ** L.Lesniak

**Classif.: ** * 05C38 Paths and cycles

**Keywords: ** subgraph of minimum degree; cycle lengths; proper subgraph

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