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

Abstract

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.