The inducibility of blow-up graphs

Abstract

Abstract. The blow-up of a graph is obtained by replacing every vertex with a finite collection of copies so that the copies of two vertices are adjacent if and only if the originals are. If every vertex is replaced with the same number of copies, then the resulting graph is called a balanced blow-up.We show that any graph which contains the maximum number of induced copies of a sufficiently large balanced blow-up of H is itself essentially a blow-up of H. This gives an asymptotic answer to a question in [BEHJ95].

Strong forms of stability from flag algebra calculations

“…G 20 x 1 To prove 1, we assume, on the contrary, that V (1,1,1) is non-empty. Thus V (1,1,1) contains at least four elements.…”

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].…”

The Inducibility of Graphs on Four Vertices

“…It is not difficult to see that this limit always exists. The very recent work gives some strong asymptotic results for a large class of graphs, but the problem of determining appears to be nontrivial even in some cases when H is a very small graph. Let denote the complement graph of H .…”

“…C 2013 Wiley Periodicals, Inc. J. Graph Theory 75: [231][232][233][234][235][236][237][238][239][240][241][242][243] 2014 It is not difficult to see that this limit always exists. The very recent work [7] gives some strong asymptotic results for a large class of graphs, but the problem of determining i(H ) appears to be nontrivial even in some cases when H is a very small graph. Let H denote the complement graph of H. Note that we have that i(H ) = i(H ), so that we need only to consider one graph in a complementary pair.…”