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Excluding Minors in Nonplanar Graphs of Girth at Least Five

Published online by Cambridge University Press:  09 April 2001

ROBIN THOMAS  and
JAN McDONALD THOMSON
ROBIN THOMAS
Affiliation:
School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia, GA 30332-0160, USA (e-mail: thomas@math.gatech.edu, thomson@math.gatech.edu)
JAN McDONALD THOMSON
Affiliation:
School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia, GA 30332-0160, USA (e-mail: thomas@math.gatech.edu, thomson@math.gatech.edu)
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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 [mid ]C[mid ] [ges ] 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.

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 9 , Issue 6 , November 2000 , pp. 573 - 585
Copyright
2000 Cambridge University Press

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