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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
cycle; triangle
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