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A long standing open problem in extremal graph theory is to describe all graphs that maximize the number of induced copies of a path on four vertices. The character of the problem changes in the setting of oriented graphs, and becomes more tractable. Here we resolve this problem in the setting of oriented graphs without transitive triangles.
The enumeration of lattice paths in wedges poses unique mathematical challenges. These models are not translationally invariant, and the absence of this symmetry complicates both the derivation of a functional recurrence for the generating function,
We determine the inducibility of all tournaments with at most $4$ vertices together with the extremal constructions. The $4$-vertex tournament containing an oriented $C_3$ and one source vertex has a particularly interesting extremal construction. It
A graph $F$ is called a fractalizer if for all $n$ the only graphs which maximize the number of induced copies of $F$ on $n$ vertices are the balanced iterated blow ups of $F$. While the net graph is not a fractalizer, we show that the net is nearly
We present a sufficient condition for the stability property of extremal graph problems that can be solved via Zykovs symmetrisation. Our criterion is stated in terms of an analytic limit version of the problem. We show that, for example, it applies
Directed paths have been used extensively in the scientific literature as a model of a linear polymer. Such paths models in particular the conformational entropy of a linear polymer and the effects it has on the free energy. These directed models are simplifi