Edge condition for chorded pancyclicity in balanced k-partite graphs

Let k and n be two integers with 𝑘 ≥ 2 and 𝑛 ≥ 1 . Let H be a balanced k -partite graph of order kn and size at least ( 𝑘 2 ) ⁢ 𝑛 2 − ( 𝑘 − 1 ) ⁢ 𝑛 + 2 . In 1988, Entringer and Schmeichel proved that H is bipancyclic for 𝑘 = 2 . In 2018, Ferrero and Lesniak showed that H is hamiltonian, and furthermore it is chorded pancyclic for 𝑘 ≥ 3 . Recently, Wang et al. showed that H is also chorded bipancyclic for 𝑘 = 2 . Note that minimum degree at least two is a necessary condition for a graph to be hamiltonian. In this paper, we let G be a balanced k -partite graph of order kn and size at least ( 𝑘 2 ) ⁢ 𝑛 2 − ( 𝑘 − 1 ) ⁢ 𝑛 + 1 with 𝛿 ⁡ ( 𝐺 ) ≥ 2 . We prove that G is hamiltonian unless G is isomorphic to one of two exceptional graphs. Furthermore, we also prove that when 𝑘 = 2 , G is chorded bipancyclic, and when 𝑘 ≥ 3 , G is chorded pancyclic unless G is isomorphic to one of four exceptional graphs.