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Hamiltonicity and related properties in $K_{r+1}$-free graphs

Hamiltonicity and related properties in $K_{r+1}$-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, $k$-path Hamiltonicity, and $k$-Hamiltonicity in $K_{r+1}$-free graphs. The problem of determining the extremal number ex$(n,F)$, the maximum number of edges in an $n$-vertex, $F$-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$(n,F)$ in which we require that the graphs not have some Hamiltonian-like property, and we determine their values for $F=K_{r+1}$. We then extend these results to clique density conditions. This talk is based on joint work with Rachel Kirsch.

## Department of Mathematics and Statistics