Min-max relations for odd cycles in planar graphs

TL;DR

This paper establishes a tighter bound on the relationship between the maximum number of vertex-disjoint odd cycles and the minimum vertex set needed to make a planar graph bipartite, improving previous bounds.

Contribution

It proves that for planar graphs, the minimum vertex removal set is at most six times the maximum number of disjoint odd cycles, refining earlier bounds.

Findings

t(G) <= 6m(G) for planar graphs

Improves previous bound of t(G) <= 10m(G)

Advances understanding of odd cycle structure in planar graphs

Abstract

Let m(G) be the maximum number of vertex-disjoint odd cycles of a graph G and t(G) the minimum number of vertices whose removal makes G bipartite. We show that t(G)<=6m(G) if G is planar. This improves the previous bound t(G)<=10m(G) by Fiorini, Hardy, Reed and Vetta [Math. Program. Ser. B 110 (2007), 71-91].