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On structure of some plane graphs with application to choosability. (English) Zbl 1024.05049
Summary: A graph \(G=(V, E)\) is \((x, y)\)-choosable for integers \(x> y\geq 1\) if for any given family \(\{A(v)\mid v\in V\}\) of sets \(A(v)\) of cardinality \(x\), there exists a collection \(\{B(v)\mid v\in V\}\) of subsets \(B(v)\subset A(v)\) of cardinality \(y\) such that \(B(u)\cap B(v)= \varnothing\) whenever \(uv\in E(G)\). In this paper, structures of some plane graphs, including plane graphs with minimum degree 4, are studied. Using these results, we may show that if \(G\) is free of \(k\)-cycles for some \(k\in \{3,4, 5,6\}\), or if any two triangles in \(G\) have distance at least 2, then \(G\) is \((4m, m)\)-choosable for all nonnegative integers \(m\). When \(m= 1\), \((4m, m)\)-choosable is simply 4-choosable. So these conditions are also sufficient for a plane graph to be 4-choosable.
Reviewer: Reviewer (Berlin)

MSC:
05C38 Paths and cycles
05C10 Planar graphs; geometric and topological aspects of graph theory
05C75 Structural characterization of families of graphs
Keywords:
cycle; triangle
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