Unavoidable parallel minors of 4-connected graphs

Abstract

A parallel minor is obtained from a graph by any sequence of edge contractions and parallel edge deletions. We prove that, for any positive integer k, every internally 4-connected graph of sufficiently high order contains a parallel minor isomorphic to a variation of K4,kwith a complete graph on the vertices of degree k, the k-partition triple fan with a complete graph on the vertices of degree k, the k-spoke double wheel, the k-spoke double wheel with axle, the (2k+1)-rung Mö bius zigzag ladder, the (2k)- rung zigzag ladder, or Kk.We also find the unavoidable parallel minors of 1-, 2-, and 3-connected graphs. © 2008 Wiley Periodicals, Inc.