On triangle-free graphs maximizing embeddings of bipartite graphs

In 1991 Gy\H ori, Pach, and Simonovits proved that for any bipartite graph $H$ containing a matching avoiding at most 1 vertex, the maximum number of copies of $H$ in any large enough triangle-free graph is achieved in a balanced complete bipartite graph. In this paper we improve their result by showing that if $H$ is a bipartite graph containing a matching of size $x$ and at most $\frac{1}{2}\sqrt{x-1}$ unmatched vertices, then the maximum number of copies of $H$ in any large enough triangle-free graph is achieved in a complete bipartite graph. We also prove that such a statement cannot hold if the number of unmatched vertices is $\Omega(x)$

Everything you always wanted to know about the parameterized complexity of Subgraph Isomorphism (but were afraid to ask)

Given two graphs H and G, the Subgraph Isomorphism problem asks if H is isomorphic to a subgraph of G. While NP-hard in general, algorithms exist for various parameterized versions of the problem. However, the literature contains very little guidance on which combinations of parameters can or cannot be exploited algorithmically. Our goal is to systematically investigate the possible parameterized algorithms that can exist for Subgraph Isomorphism. We develop a framework involving 10 relevant parameters for each of H and G (such as treewidth, pathwidth, genus, maximum degree, number of vertices, number of components, etc.), and ask if an algorithm with running time f1_(p_1,p_2,...,p_l).n^f_2(p_(l+1),...,p_k) exists, where each of p_1,...,p_k is one of the 10 parameters depending only on H or G. We show that all the questions arising in this framework are answered by a set of 11 maximal positive results (algorithms) and a set of 17 maximal negative results (hardness proofs); some of these results already appear in the literature, while others are new in this paper. On the algorithmic side, our study reveals for example that an unexpected combination of bounded degree, genus, and feedback vertex set number of G gives rise to a highly nontrivial algorithm for Subgraph Isomorphism. On the hardness side, we present W[1]-hardness proofs under extremely restricted conditions, such as when H is a bounded-degree tree of constant pathwidth and G is a planar graph of bounded pathwidth

On generalized Tur\'an problems with bounded matching number

Given a graph $H$ and a family of graphs $\mathcal{F}$, the generalized Tur\'an number $\mathrm{ex}(n,H,\mathcal{F})$ is the maximum number of copies of $H$ in an $n$-vertex graphs that do not contain any member of $\mathcal{F}$ as a subgraph. Recently there has been interest in studying the case $\mathcal{F}=\{F,M_{s+1}\}$ for arbitrary $F$ and $H=K_r$. We extend these investigations to the case $H$ is arbitrary as well

Generalized Tur\'an results for disjoint cliques

The generalized Tur\'an number $\mathrm{ex}(n,H,F)$ is the largest number of copies of $H$ in $n$-vertex $F$-free graphs. We denote by $tF$ the vertex-disjoint union of $t$ copies of $F$. Gerbner, Methuku and Vizer in 2019 determined the order of magnitude of $\mathrm{ex}(n,K_s,tK_r)$. We extend this result in three directions. First, we determine $\mathrm{ex}(n,K_s,tK_r)$ exactly for sufficiently large $n$. Second, we determine the asymptotics of the analogous number for $p$-uniform hypergraphs. Third, we determine the order of magnitude of $\mathrm{ex}(n,H,tK_r)$ for every graph $H$, and also of the analogous number for $p$-uniform hypergraphs.Comment: 10 page

Some exact results for non-degenerate generalized Tur\'an problems

The generalized Tur\'an number $\mathrm{ex}(n,H,F)$ is the maximum number of copies of $H$ in $n$-vertex $F$-free graphs. We consider the case where $\chi(H)<\chi(F)$. There are several exact results on $\mathrm{ex}(n,H,F)$ when the extremal graph is a complete $(\chi(F)-1)$-partite graph. We obtain multiple exact results with other kinds of extremal graphs

Generalized regular Tur\'an numbers

We combine two generalizations of ordinary Tur\'an problems. Given graphs $H$ and $F$ and a positive integer $n$, we study $rex(n, H, F )$, which is the largest number of copies of $H$ in $F$-free regular $n$-vertex graphs.Comment: 13 page

The Extremal Function for K10 Minors

We prove that every graph on n >= 8 vertices and at least 8n-35 edges either has a K10 minor or is isomorphic to some graph included in a few families of exceptional graphs.Ph.D