an:01843773
Zbl 1024.05049
Lam, Peter Che Bor; Shiu, Wai Chee; Xu, Baogang
On structure of some plane graphs with application to choosability
EN
J. Comb. Theory, Ser. B 82, No. 2, 285-296 (2001).
0095-8956
2001
j
05C38 05C10 05C75
cycle; triangle
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.