## Polytopic locally linear graphs.(English)Zbl 0642.05029

Denote by $$N_ G(v)$$ the subgraph of the graph G induced by the set of vertices adjacent to v. If $$N_ G(v)$$ is regular of degree 1 for all vertices v of G, then G is called locally linear. These graphs were introduced by D. Fronček at the Czech. Conf. Graph Theory in the Raček Valley in May 1986. Here attention is restricted to such graphs that are also planar and 3-connected. Sharp bounds for the number of edges in these graphs are obtained and the extremal graphs are examined.
Reviewer: R.C.Entringer

### MSC:

 05C35 Extremal problems in graph theory
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### References:

 [1] FRONČEK D.: Locally linear graphs. (Slovak.) Lecture at the Czechoslovak Conference on Graph Theory in the Raček Valley in May 1986. [2] SEDLÁČEK J.: Local properties of graphs. (Czech.) Časop. pěst. mat. 106, 1981, 290-298. · Zbl 0478.05080 [3] SEDLÁČEK J.: On local properties of finite graphs. Graph Theory, Lagow 1981 (ed. by M. Borowiecki, J. W. Kennedy and M. M. Syslo). Springer Verlag Berlin-Heidelberg-New York-Tokyo 1983.
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