\([2,3]\)-factors in a 3-connected infinite planar graph.
Let \(G = (V,E)\) be a graph. An \([m,n]\)-factor in \(G\) is a spanning subgraph \(F\) of \(G\) which satisfies the property: \(m \leq d_{F}(x) \leq n,\) for all \(x \in V(F).\) The author investigates \([m,n]\)-factors of 3-connected locally finite infinite planar graphs containing no unbounded faces. Let \(G\) be such a graph with additional conditions (i) it contains no vertex accumulation points; and (ii) \(d_{G}(v) \geq 4\) for all \(v \in V(G).\) The main result is the theorem that for every such graph there exists a \([2,3]\)-factor. This theorem extends the result by \textit{H. Enomoto, T. Iida} and \textit{K. Ota} [J. Comb. Theory, Ser. B 68, 314--323 (1996; Zbl 0861.05022)] that every 3-connected planar graph whose minimum degree is greater than or equal to 4 has a connected \([2,3]\)-factor. The organization of the paper is as follows: 1. Introduction. 2. Preliminary results. 3. Factors in a bridge. 4. Suitable factors and proof of the main theorem. In concluding remarks the author formulates the conjecture that every 3-connected locally finite infinite planar graph satisfying \(d_{G}(v) \geq 4\) for all \(v \in V(G)\) contains a \([2,3]\)-factor.
scientific article; zbMATH DE number 1836013
scientific article