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Do 3 n − 5 edges force a subdivision of K 5 ?
Author(s) -
Kézdy André E.,
McGuinness Patrick J.
Publication year - 1991
Publication title -
journal of graph theory
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 54
eISSN - 1097-0118
pISSN - 0364-9024
DOI - 10.1002/jgt.3190150405
Subject(s) - counterexample , subdivision , mathematics , combinatorics , conjecture , graph , simple (philosophy) , dirac (video compression format) , euler characteristic , discrete mathematics , physics , philosophy , archaeology , epistemology , nuclear physics , neutrino , history
A conjecture of Dirac states that every simple graph with n vertices and 3 n − 5 edges must contain a subdivision of K 5 . We prove that a topologically minimal counterexample is 5‐connected, and that no minor‐minimal counterexample contains K 4 – e . Consequently, Dirac's conjecture holds for all graphs that can be embedded in a surface with Euler characteristic at least − 2.
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