Related papers: Graphs with many copies of a given subgraph

A Hamiltonian graph $G$ of order $n$ is $k$-ordered, $2\leq k \leq n$, if for every sequence $v_1, v_2, \ldots ,v_k$ of $k$ distinct vertices of $G$, there exists a Hamiltonian cycle that encounters $v_1, v_2, \ldots , v_k$ in this order.…

We prove that if an $n$-vertex graph $G$ can be drawn in the plane such that each pair of crossing edges is independent and there is a crossing-free edge that connects their endpoints, then $G$ has $O(n)$ edges. Graphs that admit such…

Let $G$ be a simple graph on $n$ vertices. Let $H$ be either the complete graph $K_m$ or the complete bipartite graph $K_{r,s}$ on a subset of the vertices in $G$. We show that $G$ contains $H$ as a subgraph if and only if…

We say that a graph G has a perfect H-packing (also called an H-factor) if there exists a set of disjoint copies of H in G which together cover all the vertices of G. Given a graph H, we determine, asymptotically, the Ore-type degree…

In this note, we fix a graph $H$ and ask into how many vertices can each vertex of a clique of size $n$ can be "split" such that the resulting graph is $H$-free. Formally: A graph is an $(n,k)$-graph if its vertex sets is a pairwise…

We show that if pn >> log n, the binomial random graph G_{n,p} has an approximate Hamilton decomposition. More precisely, we show that in this range G_{n,p} contains a set of edge-disjoint Hamilton cycles covering almost all of its edges.…

The inducibility of a graph $H$ measures the maximum number of induced copies of $H$ a large graph $G$ can have. Generalizing this notion, we study how many induced subgraphs of fixed order $k$ and size $\ell$ a large graph $G$ on $n$…

We say that a graph $H$ dominates another graph $H'$ if the number of homomorphisms from $H'$ to any graph $G$ is dominated, in an appropriate sense, by the number of homomorphisms from $H$ to $G$. We study the family of dominating graphs,…

Let $C(n)$ denote the maximum number of induced copies of 5-cycles in graphs on $n$ vertices. For $n$ large enough, we show that $C(n)=a\cdot b\cdot c \cdot d \cdot e + C(a)+C(b)+C(c)+C(d)+C(e)$, where $a+b+c+d+e = n$ and $a,b,c,d,e$ are as…

We prove a version of the Loebl-Komlos-Sos Conjecture for dense graphs. For each q>0 there exists a number $n_0\in \mathbb{N}$ such that for any n>n_0 and k>qn the following holds: if G be a graph of order n with at least n/2 vertices of…

Let $I(n)$ denote the number of isomorphism classes of subgroups of $(\Bbb Z/n\Bbb Z)^\times$, and let $G(n)$ denote the number of subgroups of $(\Bbb Z/n\Bbb Z)^\times$ counted as sets (not up to isomorphism). We prove that both $\log…

We consider three extremal problems about the number of copies of a fixed graph in another larger graph. First, we correct an error in a result of Reiher and Wagner and prove that the number of $k$-edge stars in a graph with density $x \in…

In this article we study graphs with ordering of vertices, we define a generalization called a pseudoordering, and for a graph $H$ we define the $H$-Hamiltonian number of a graph $G$. We will show that this concept is a generalization of…

Call a simple graph $H$ of order $n$ well-separable, if by deleting a separator set of size $o(n)$ the leftover will have components of size at most $o(n)$. We prove, that bounded degree well-separable spanning subgraphs are easy to embed:…

A simple topological graph T = (V(T), E(T)) is a drawing of a graph in the plane where every two edges have at most one common point (an endpoint or a crossing) and no three edges pass through a single crossing. Topological graphs G and H…

Intuitively, a tangle of large order in a graph is a highly-connected part of the graph, and it is known that if a graph has a tangle of large order then it has a large grid minor. Here we show that for any k, if G has a tangle of large…

By a poly-line drawing of a graph G on n vertices we understand a drawing of G in the plane such that each edge is represented by a polygonal arc joining its two respective vertices. We call a turning point of a polygonal arc the bend. We…

We shall present an algorithm for determining whether or not a given planar graph H can ever be a subgraph of a 4-regular planar graph. The algorithm has running time O(|H|^{2.5}) and can be used to find an explicit 4-regular planar graph G…

A graph $G$ of order $2n$ is called degree-equipartite if for every $n$-element set $A\subseteq V(G)$, the degree sequences of the induced subgraphs $G[A]$ and $G[V(G)\setminus A]$ are the same. In this paper, we characterize all…

Let $H$ and $G$ be graphs such that $H$ has at least 3 vertices and is connected. The $H$-line graph of $G$, denoted by $HL(G)$, is that graph whose vertices are the edges of $G$ and where two vertices of $HL(G)$ are adjacent if they are…