1 research outputs found
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
and a hypercube , is the largest number of edges in an
-free subgraph of . If is equal to a positive proportion
of the number of edges in , is said to have positive Tur\'an density
in a hypercube; otherwise it has zero Tur\'an density. Determining
and even identifying whether has positive or zero Tur\'an density remains a
widely open question for general .
In this paper we focus on layered graphs, i.e., graphs that are contained in
an edge-layer of some hypercube. Graphs that are not layered have positive
Tur\'an density because one can form an -free subgraph of consisting
of edges of every other layer. For example, a -cycle is not layered and has
positive Tur\'an density.
However, in general it is not obvious what properties layered graphs have. We
give a characterisation of layered graphs in terms of edge-colorings and show
that any -vertex layered graphs has at most
edges. We show that most non-trivial subdivisions have zero Tur\'an density,
extending known results on zero Tur\'an density of even cycles of length at
least and of length . However, we prove that there are cubical graphs
of girth that are not layered and thus having positive Tur\'an density. The
cycle of length remains the only cycle for which it is not known whether
its Tur\'an density is positive or not. We prove that , for a constant , showing that the extremal number
for a -cycle behaves differently from any other cycle of zero Tur\'an
density