The Representation of a Graph by Set Intersections

Abstract

Geometrically, a graph is a collection of points (or vertices) together with a set of edges (or curves) each of which joins two distinct vertices of the graph, and no two of which have points in common except possibly end points. Two given vertices of the graph may be joined by no edge or one edge, but may not be joined by more than one edge. From an abstract point of view, a graph G is a collection of elements ﹛x1, x2, …﹜ called points or vertices, together with a second collection of certain pairs (xα, Xβ) of distinct points of G. It is helpful to retain the geometric language, and refer to any pair in as an edge (or a curve) of G that joins the points xα and Xβ.

Extremal clique coverings of complementary graphs

“…In [2] it is shown that cc(D)-k2/4 for all k-vertex graphs D . Therefore the edges of H with both ends in U or both ends in V can be covered by at most IUJ 2 /4 or JV1 2/4 cliques respectively .…”

Minimum rainbow H-decompositions of graphs

“…They showed that φ(n, K 3 ) = ex(n, K 3 ), where ex(n, H) denotes the maximum size of a graph on n vertices, that does not contain H as a subgraph. Moreover, these authors proved in [5] that the only graph that maximizes this function is the complete balanced bipartite graph. Consequently, they conjectured that φ(n, K r ) = ex(n, K r ) and the only optimal graph is the Turán graph T r−1 (n), the complete balanced (r − 1)-partite graph on n vertices, where the sizes of the partite sets differ from each other by at most one.…”

“…This function was studied first by Erdős, Goodman and Pósa [5], who were motivated by the problem of representing graphs by set intersections. They showed that φ(n, K 3 ) = ex(n, K 3 ), where ex(n, H) denotes the maximum size of a graph on n vertices, that does not contain H as a subgraph.…”

On set intersection representations of graphs

“…-A classical result of Erdős, Goodman and Pósa [16] states that Θ 1 (G) ≤ Θ 1 (K n,n ) = n 2 for every graph G on 2n vertices, where K n,n is a complete bipartite n × n graph. Moreover, a desired covering of G can be achieved by only using simplest complete subgraphs: edges and triangles.…”