A Note on the Inducibility of $$4$$ 4 -Vertex Graphs

Abstract

There is much recent interest in understanding the density at which constant size graphs can appear in a very large graph. Specifically, the inducibility of a graph H is its extremal density, as an induced subgraph of G, where |G| → ∞. Already for 4-vertex graphs many questions are still open. Thus, the inducibility of the 4-path was addressed in a construction of Exoo (Ars Combin 22:5-10, 1986), but remains unknown. Refuting a conjecture of Erdős, Thomason (Combinatorica 17(1):125-134, 1997) constructed graphs with a small density of both 4-cliques and 4-anticliques. In this note, we merge these two approaches and construct better graphs for both problems.

Inducibility of d-ary trees

“…In 2014, James Hirst [14] determined, employing Razborov's flag algebra method and semi-definite programming techniques, the inducibility of two 4-vertex graphs: the complete tripartite graph K 1,1,2 and the so-called paw graph (graph constructed from a triangle by appending a pendant edge). The concept of inducibility is still gaining consideration from several research groups; see [13] and [9] for some recent results on blow-up of graphs and graphs on four vertices, respectively. The language of flag algebra was also employed recently in [1] to derive the inducibility of the cycle on five vertices, thereby settling a particular case of a conjecture formulated in [16].…”

Maximum density of induced 5-cycle is achieved by an iterated blow-up of 5-cycle

“…Problems of maximizing the number of induced copies of a fixed small graph H have attracted a lot of attention recently [8,14,29]. For a list of other results on this so called inducibility of small graphs of order up to 5, see the work of Even-Zohar and Linial [8].…”

Inducibility in Binary Trees and Crossings in Random Tanglegrams

“…is called the inducibility of G. The concept is still investigated vigorously to this day, see [4,6] for some recent results.…”