Related papers: Completing Partial Packings of Bipartite Graphs

Let $H$ be an $n$-vertex 3-uniform hypergraph such that every pair of vertices is in at least $n/3+o(n)$ edges. We show that $H$ contains two vertex-disjoint tight paths whose union covers the vertex set of $H$. The quantity two here is…

We study graph partitioning problems from a min-max perspective, in which an input graph on n vertices should be partitioned into k parts, and the objective is to minimize the maximum number of edges leaving a single part. The two main…

In this paper we provide a method of finding possible numbers of shortest paths between two points in a space of compact sets in Euclidean space with Hausdorff distance. We also prove that there cannot be some of the numbers of shortest…

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…

One of the most basic questions one can ask about a graph $H$ is: how many $H$-free graphs on $n$ vertices are there? For non-bipartite $H$, the answer to this question has been well-understood since 1986, when Erd\H{o}s, Frankl and R\"odl…

We present a tight extremal threshold for the existence of Hamilton cycles in graphs with large minimum degree and without a large ``bipartite hole`` (two disjoint sets of vertices with no edges between them). This result extends Dirac's…

Recently, Letzter proved that any graph of order $n$ contains a collection $\mathcal{P}$ of $O(n\log^\star n)$ paths with the following property: for all distinct edges $e$ and $f$ there exists a path in $\mathcal{P}$ which contains $e$ but…

We consider the following problem: let $n>k$ be natural numbers, and let $G$ be a graph on $n$ vertices (undirected, without loops or multiple edges). Denote by $h_k(G)$ the number of unordered pairs of vertices in the graph $G$ whose…

Let $G$ be a graph on $n$ vertices and $(H,+)$ be an abelian group. What is the minimum size ${\sf S}_H(G)$ of the set of all sums $A(u)+A(v)$ over all injections $A:V(G)\to H$? In 2012, the first author, Angel, the second author, and…

Given a family $\mathcal{F}$ of bipartite graphs, the {\it Zarankiewicz number} $z(m,n,\mathcal{F})$ is the maximum number of edges in an $m$ by $n$ bipartite graph $G$ that does not contain any member of $\mathcal{F}$ as a subgraph (such…

One of the fundamental results in graph minor theory is that for every planar graph $H$, there is a minimum integer $f(H)$ such that graphs with no minor isomorphic to $H$ have treewidth at most $f(H)$. A lower bound for ${f(H)}$ can be…

For integers $k\geq 1$ and $n\geq 2k+1$ the Kneser graph $K(n,k)$ has as vertices all $k$-element subsets of $[n]:=\{1,2,\ldots,n\}$ and an edge between any two vertices (=sets) that are disjoint. The bipartite Kneser graph $H(n,k)$ has as…

For a bipartite graph G, let h(G) be the largest t such that either G or the bipartite complement of G contain K_{t,t}. For a class F of graphs, let h(F)= min {h(G): G\in F}. We say that a bipartite graph H is strongly acyclic if neither H…

Given a real $\mu\geq 1$, a graph $H$ is $\mu$-almost-regular if $\Delta(H)\leq \mu \delta(H)$. The celebrated regularization theorem of Erd\H{o}s and Simonovits states that for every real $0<\varepsilon<1$ there exists a real…

A maximum weighted matching for bipartite graphs $G=(A \cup B,E)$ can be found by using the algorithm of Edmonds and Karp with a Fibonacci Heap and a modified Dijkstra in $O(nm + n^2 \log{n})$ time where n is the number of nodes and m the…

We study the problem of determining $sat(n,k,r)$, the minimum number of edges in a $k$-partite graph $G$ with $n$ vertices in each part such that $G$ is $K_r$-free but the addition of an edge joining any two non-adjacent vertices from…

For an odd integer $k$, let $\mathcal{C}_k = \{C_3,C_5,...,C_k\}$ denote the family of all odd cycles of length at most $k$ and let $\mathcal{C}$ denote the family of all odd cycles. Erd\H{o}s and Simonovits \cite{ESi1} conjectured that for…

The supersaturation problem for a given graph $F$ asks for the minimum number $h_F(n,q)$ of copies of $F$ in an $n$-vertex graph with $ex(n,F)+q$ edges. Subsequent works by Rademacher, Erd\H{o}s, and Lov\'{a}sz and Simonovits determine the…

Fix $\varepsilon >0$ and consider a multipartite graph $G$ with maximum degree at most $(1-\varepsilon)n$, parts $V_1,\ldots,V_k$ of the same size $n$, and where every vertex has at most $o(n)$ neighbors in any part $V_i$. Loh and Sudakov…

Erd\H{o}s and Hajnal proposed a problem that: is it true that every $(2n+1)$-vertex graph with $n^2+n+1$ edges contains two vertices of equal degree connected by a path of length three? The edge bound is sharp by the complete bipartite…