Excluding Minors in Nonplanar Graphs of Girth at Least Five

Authors:
Robin Thomas;Jan McDonald Thomson
Affiliations:
School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia, GA 30332-0160, USA (e-mail: thomas@math.gatech.edu, thomson@math.gatech.edu);School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia, GA 30332-0160, USA (e-mail: thomas@math.gatech.edu, thomson@math.gatech.edu)
Venue:
Combinatorics, Probability and Computing
Year:
2000

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Abstract

A graph G is quasi 4-connected if it is simple, 3-connected, has at least five vertices, and for every partition (A, B, C) of V(G) either ∣C∣ ≥ 4, or G has an edge with one end in A and the other end in B, or one of A,B has at most one vertex. We show that any quasi 4-connected nonplanar graph with minimum degree at least three and no cycle of length less than five has a minor isomorphic to P−10, the Petersen graph with one edge deleted. We deduce the following weakening of Tutte's Four Flow Conjecture: every 2-edge-connected graph with no minor isomorphic to P−10 has a nowhere-zero 4-flow. This extends a result of Kilakos and Shepherd who proved the same for 3-regular graphs.