Trapezoid graphs are intersection graphs of trapezoids between two horizontal lines. They generalize both interval graphs and permutation graphs. Interval graphs are intersection graphs of intervals on the real line. Permutation graphs are intersection graphs of straight lines between two horizontal lines. They can be represented by a permutation diagram consisting of a pair of horizontal lines...

We collect some general results on graph limits associated to hereditary classes of graphs. As examples, we consider some classes defined by forbidden subgraphs and some classes of intersection graphs, including triangle-free graphs, chordal graphs, cographs, interval graphs, unit interval graphs, threshold graphs, and line graphs.

Let x(G) and w(G) denote the chromatic number and clique number (maximum size of a clique) of a graph G. To avoid trivial cases, we always assume that w (G);?: 2. It is well known that interval graphs are perfect, in particular x( G)= w (G) for every interval graph G. In this paper we study the closeness of x and w for two well-known non-perfect relatives of interval graphs: multiple interval g...

Tolerance graphs have been extensively studied since their introduction in 1982 [9], due to their interesting structure and their numerous applications, as they generalize both interval and permutation graphs in a natural way. It has been conjectured in 1984 that the intersection of tolerance and cocomparability graphs coincides with bounded tolerance graphs [10]. The conjecture has been proved...

Ptolemaic graphs are graphs that satisfy the Ptolemaic inequality for any four vertices. The graph class coincides with the intersection of chordal graphs and distance hereditary graphs, and it is a natural generalization of block graphs (and hence trees). In this paper, a new characterization of ptolemaic graphs is presented. It is a laminar structure of cliques, and leads us to a canonical tr...

In this paper, we study the Hankel edge ideals of graphs. We determine minimal prime ideal labeled Hamiltonian and semi-Hamiltonian graphs, investigate radicality, being a complete intersection, almost intersection set-theoretic for such also consider trees with natural labeling, called rooted labeling. characterize whose is moreover, those initial respect to reverse lexicographic order satisfi...