Nearly bipartite graphs
We prove that if a nonbipartite graph G on n vertices has minimal degree δ≥n/(4k+2)+ck,m, where ck,m does not depend on n and n is sufficiently large, if C2s+1⊂G for some k≤s≤4k+1 then C2s+2j+1⊂G for every j=1,...,m. We give a structural description of all graphs on n vertices with δ≥n/(4k+2) and not containing odd cycles of order larger than 2k+1 and show that they can be made bipartite by deletion of a fixed number of edges or vertices. Such graphs will be called nearly bipartite graphs. © 2003 Elsevier B.V. All rights reserved.
Gyori, E., Nikiforov, V., & Schelp, R. (2003). Nearly bipartite graphs. Discrete Mathematics, 272 (2022-02-03), 187-196. https://doi.org/10.1016/S0012-365X(03)00076-1