an:01970687
Zbl 1023.05045
Alon, Noga; Ding, Guoli; Oporowski, Bogdan; Vertigan, Dirk
Partitioning into graphs with only small components
EN
J. Comb. Theory, Ser. B 87, No. 2, 231-243 (2003).
00098911
2003
j
05C15 05C70
tree-width; vertex partition; edge partition; graph coloring
The authors prove several results on edge partitions and vertex partitions of graphs into graphs with bounded size components. The main results are:
(1) Every graph with maximum degree at most \(\Delta\) and tree-width at most \(k\) admits a vertex partition into two induced subgraphs \(G_1\), \(G_2\) such that each connected component of \(G_1\) and \(G_2\) has at most \(24k\Delta\) vertices, and an edge partition into two subgraphs \(H_1\), \(H_2\) such that each connected component of \(H_1\) and \(H_2\) has at most \(24k\Delta(\Delta+1)\) vertices.
(2) Every graph with maximum degree \(\Delta\geq 3\) admits a vertex partition into \(\lfloor\frac{\Delta+2}{3}\rfloor\) induced subgraphs \(G_i\) such that each connected component of \(G_i\) has at most \(12\Delta^2-36\Delta+9\) vertices.
(3) Every graph with maximum degree \(\Delta\geq 2\) admits an edge partition into \(\lfloor\frac{\Delta+1}{2}\rfloor\) subgraphs \(H_i\) such that each connected component of \(H_i\) has at most \(60\Delta-63\) edges.
(4) For every integer \(n\), there is a planar graph of maximum degree six such that in every vertex partition and every edge partition \(\{G_1, G_2\}\), one of \(G_1\), \(G_2\) must have a connected component with at least \(n\) vertices, and there is a planar graph such that in every vertex partition \(\{G_1, G_2, G_3\}\), one of \(G_1\), \(G_2\), \(G_3\) must have a connected component with at least \(n\) vertices.
Van Bang Le (Rostock)