Packing and covering odd cycles in cubic plane graphs with small faces

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Abstract

We show that any 3-connected cubic plane graph on n vertices, with all faces of size at most 6, can be made bipartite by deleting no more than (p+3t)n/5 edges, where p and t are the numbers of pentagonal and triangular faces, respectively. In particular, any such graph can be made bipartite by deleting at most 12n/5 edges. This bound is tight, and we characterise the extremal graphs. We deduce tight lower bounds on the size of a maximum cut and a maximum independent set for this class of graphs. This extends and sharpens the results of Faria, Klein and Stehlík [SIAM J. Discrete Math. 26 (2012) 1458–1469].

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1

Partially supported by CAPES and CNPq.

2

Partially supported by ANR project Stint (ANR-13-BS02-0007), ANR project GATO (ANR-16-CE40-0009-01), and by Labex PERSYVAL-Lab (ANR-11-LABX-0025).

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