On Graphs Embeddable in a Layer of a Hypercube and Their Extremal Numbers

A graph is cubical if it is a subgraph of a hypercube. For a cubical graph H and a hypercube $Q_n$ , $\text{ex}(Q_n,H)$ is the largest number of edges in an H-free subgraph of $Q_n$ . If $\text{ex}(Q_n,H)$ is at least a positive proportion of the number of edges in $Q_n$ , then H is said to have positive Turán density in the hypercube; otherwise it has zero Turán density. Determining $\text{ex}(Q_n,H)$ and even identifying whether H has positive or zero Turán density remains a widely open question for general H. In this paper we focus on layered graphs, i.e., graphs that are contained in an edge layer of some hypercube. Graphs H that are not layered have positive Turán density because one can form an H-free subgraph of $Q_n$ consisting of edges of every other layer. For example, a 4-cycle is not layered and has positive Turán density. However, in general, it is not obvious what properties layered graphs have. We give a characterization of layered graphs in terms of edge-colorings. We show that most non-trivial subdivisions have zero Turán density, extending known results on zero Turán density of even cycles of length at least 12 and of length 8. However, we prove that there are cubical graphs of girth 8 that are not layered and thus having positive Turán density. The cycle of length 10 remains the only cycle for which it is not known whether its Turán density is positive or not. We prove that $\text{ex}(Q_n,C_{10})=\Omega (n{2}^n/{log}^{a}n)$ , for a constant a, showing that the extremal number for a 10-cycle behaves differently from any other cycle of zero Turán density.