Hamiltonicity of edge chromatic critical graphs

Vizing conjectured that every edge chromatic critical graph contains a 2-factor. Believing that stronger properties hold for this class of graphs, Luo and Zhao (2013) showed that every edge chromatic critical graph of order n with maximum degree at least 6n7 is Hamiltonian. Furthermore, Luo et al. (2016) proved that every edge chromatic critical graph of order n with maximum degree at least 4n5 is Hamiltonian. In this paper, we prove that every edge chromatic critical graph of order n with maximum degree at least 3n4 is Hamiltonian. Our approach is inspired by the recent development of Kierstead path and Tashkinov tree techniques for multigraphs.