Hamiltonicity and related properties in Kr+1 -free graphs

In this talk, we discuss a new result on best-possible edge density conditions sufficient to imply traceability, Hamiltonicity, chorded pancyclicity, Hamiltonian-connectedness, k𝑘-path Hamiltonicity, and k𝑘-Hamiltonicity in Kr+1𝐾𝑟+1-free graphs. The problem of determining the extremal number ex(n,F)(𝑛,𝐹), the maximum number of edges in an n𝑛-vertex, F𝐹-free graph, has been studied extensively since Tur\'{a}n's theorem. Edge density conditions implying these properties also had been found. We bring together these two themes. Equivalently, we introduce variants of the extremal number ex(n,F)(𝑛,𝐹) in which we require that the graphs not have some Hamiltonian-like property, and we determine their values for F=Kr+1𝐹=𝐾𝑟+1. We then extend these results to clique density conditions. This talk is based on joint work with Rachel Kirsch.