Some extremal problems for hereditary properties of graphs

This note answers extremal questions like: what is the maximum number of edges in a graph of order n, which belongs to some hereditary property. The same question is answered also for the spectral radius and other similar parameters

Minimal reducible bounds for induced-hereditary properties

AbstractLet (Ma,⊆) and (La,⊆) be the lattices of additive induced-hereditary properties of graphs and additive hereditary properties of graphs, respectively. A property R∈Ma (∈La) is called a minimal reducible bound for a property P∈Ma (∈La) if in the interval (P,R) of the lattice Ma (La) there are only irreducible properties. The set of all minimal reducible bounds of a property P∈Ma in the lattice Ma we denote by BM(P). Analogously, the set of all minimal reducible bounds of a property P∈La in La is denoted by BL(P).We establish a method to determine minimal reducible bounds for additive degenerate induced-hereditary (hereditary) properties of graphs. We show that this method can be successfully used to determine already known minimal reducible bounds for k-degenerate graphs and outerplanar graphs in the lattice La. Moreover, in terms of this method we describe the sets of minimal reducible bounds for partial k-trees and the graphs with restricted order of components in La and k-degenerate graphs in Ma

Hereditary properties of partitions, ordered graphs and ordered hypergraphs

In this paper we use the Klazar-Marcus-Tardos method to prove that if a hereditary property of partitions P has super-exponential speed, then for every k-permutation pi, P contains the partition of [2k] with parts {i, pi(i) + k}, where 1 <= i <= k. We also prove a similar jump, from exponential to factorial, in the possible speeds of monotone properties of ordered graphs, and of hereditary properties of ordered graphs not containing large complete, or complete bipartite ordered graphs. Our results generalize the Stanley-Wilf Conjecture on the number of n-permutations avoiding a fixed permutation, which was recently proved by the combined results of Klazar and of Marcus and Tardos. Our main results follow from a generalization to ordered hypergraphs of the theorem of Marcus and Tardos.Comment: 25 pgs, no figure

Hereditary properties of combinatorial structures: posets and oriented graphs

A hereditary property of combinatorial structures is a collection of structures (e.g. graphs, posets) which is closed under isomorphism, closed under taking induced substructures (e.g. induced subgraphs), and contains arbitrarily large structures. Given a property P, we write P_n for the collection of distinct (i.e., non-isomorphic) structures in a property P with n vertices, and call the function n -> |P_n| the speed (or unlabelled speed) of P. Also, we write P^n for the collection of distinct labelled structures in P with vertices labelled 1,...,n, and call the function n -> |P^n| the labelled speed of P. The possible labelled speeds of a hereditary property of graphs have been extensively studied, and the aim of this paper is to investigate the possible speeds of other combinatorial structures, namely posets and oriented graphs. More precisely, we show that (for sufficiently large n), the labelled speed of a hereditary property of posets is either 1, or exactly a polynomial, or at least 2^n - 1. We also show that there is an initial jump in the possible unlabelled speeds of hereditary properties of posets, tournaments and directed graphs, from bounded to linear speed, and give a sharp lower bound on the possible linear speeds in each case.Comment: 26 pgs, no figure

Consistent random vertex-orderings of graphs

Given a hereditary graph property $\mathcal{P}$, consider distributions of random orderings of vertices of graphs $G\in\mathcal{P}$ that are preserved under isomorphisms and under taking induced subgraphs. We show that for many properties $\mathcal{P}$ the only such random orderings are uniform, and give some examples of non-uniform orderings when they exist

Minimal reducible bounds for the class of k-degenerate graphs

AbstractLet (La,⊆) be the lattice of hereditary and additive properties of graphs. A reducible property R∈La is called minimal reducible bound for a property P∈La if in the interval (P,R) of the lattice La, there are only irreducible properties. We prove that the set B(Dk)={Dp∘Dq:k=p+q+1} is the covering set of minimal reducible bounds for the class Dk of all k-degenerate graphs