Arrangeability and Clique Subdivisions

Summary

Let k be an integer. A graph G is k-arrangeable (concept introduced by Chen and Schelp) if the vertices of G can be numbered v ₁, v ₂, …, v _n in such a way that for every integer i with 1 ≤ i ≤ n, at most k vertices among {v ₁, v ₂, …, v _i } have a neighbor $v \in \{ v_{i+1},v_{i+2},\ldots,v_{n}\}$ that is adjacent to v _i . We prove that for every integer p ≥ 1, if a graph G is not 2500(p + 1)⁸-arrangeable, then it contains a K _p -subdivision. By a result of Chen and Schelp this implies that graphs with no K _p -subdivision have “linearly bounded Ramsey numbers,” and by a result of Kierstead and Trotter it implies that such graphs have bounded “game chromatic number.”

Supported in part by NSF under Grant No. DMS-9401559.

Supported in part by NSF under Grant No. DMS-9303761, and by ONR under Contract No. N00014-93-1-0325.