Graphs with Girth 2+1 and Without Longer Odd Holes that Contain an Odd K₄-Subdivision

We say that a graph G has an odd K₄-subdivision if some subgraph of G is isomorphic to a K₄-subdivision and whose faces are all odd holes of G. For a number 2, let G_ denote the family of graphs which have girth 2+1 and have no odd hole with length greater than 2+1. Wu, Xu and Xu conjectured that every graph in ₂G_ is 3-colorable. Recently, Chudnovsky et al. and Wu et al. , respectively, proved that every graph in G₂ and G₃ is 3-colorable. In this paper, we prove that no 4-vertex-critical graph in ₅G_ has an odd K₄-subdivision. Using this result, Chen proved that all graphs in ₅G_ are 3-colorable.