Let   α 1 (G)       α    1     (  G  )         denote the maximum size of an edge set that contains at most one edge from each triangle of   G     G       . Let   τ B (G)       τ    B     (  G  )         denote the minimum size of an edge set whose deletion makes   G     G        bipartite. It was first conjectured by Lehel and later independently by Puleo that   α 1 (G)+τ B (G)≤n 2 ∕4       α    1     (  G  )   +    τ    B     (  G  )   ≤    n    2    ∕  4        for every   n     n       -vertex graph   G     G       . Puleo showed that   α 1 (G)+τ B (G)≤5n 2 ∕16       α    1     (  G  )   +    τ    B     (  G  )   ≤  5    n    2    ∕  16        for every   n     n       -vertex graph   G     G       . In this note, we improve the bound by showing that   α 1 (G)+τ B (G)≤4403n 2 ∕15000       α    1     (  G  )   +    τ    B     (  G  )   ≤  4403    n    2    ∕  15000        for every   n     n       -vertex graph   G     G       .
['Honghai Xu']
A Note on Bipartite Subgraphs and Triangle-independent Sets