Let $H$ be a fixed graph. A {\em fractional $H$-decomposition} of a graph $G$ is an assignment of nonnegative real weights to the copies of $H$ in $G$ such that for each $e \in E(G)$, the sum of the weights of copies of $H$ containing $e$ in precisely one. An {\em $H$-packing} of a graph $G$ is a set of edge disjoint copies of $H$ in $G$. The following results are proved. For every fixed $k > 2$, every graph with $n$ vertices and minimum degree at least $n(1-1/9k^{10})+o(n)$ has a fractional $K_k$-decomposition and has a $K_k$-packing which covers all but $o(n^2)$ edges.
Let ${\cal F}$ be a family of graphs. For a graph $G$, the {\em ${\cal F}$-packing number}, denoted $ν_{\cal F}(G)$, is the maximum number of pairwise edge-disjoint elements of ${\cal F}$ in $G$. A function $ψ$ from the set of elements of ${\cal F}$ in $G$ to $[0,1]$ is a {\em fractional ${\cal F}$-packing} of $G$ if $\sum_{e \in H \in {\cal F}} {ψ(H)} \leq 1$ for each $e \in E(G)$. The {\em fractional ${\cal F}$-packing number}, denoted $ν^*_{\cal F}(G)$, is defined to be the maximum value of $\sum_{H \in {{G} \choose {\cal F}}} ψ(H)$ over all fractional ${\cal F}$-packings $ψ$. Our main resu…
A well-known conjecture of Tuza asserts that if a graph has at most $t$ pairwise edge-disjoint triangles, then it can be made triangle-free by removing at most $2t$ edges. If true, the factor 2 would be best possible. In the directed setting, also asked by Tuza, the analogous statement has recently been proven, however, the factor 2 is not optimal. In this paper, we show that if an $n$-vertex directed graph has at most $t$ pairwise arc-disjoint directed triangles, then there exists a set of at most $1.8t+o(n^2)$ arcs that meets all directed triangles. We complement our result by presenting two…
We prove that there exists a positive constant εsuch that if \log n / n \le p \le n^{-1+ε}, then asymptotically almost surely the random graph G ~ G(n,p) contains a collection of \lfloor δ(G)/2 \rfloor edge-disjoint Hamilton cycles.
A set of geometric graphs is {\em geometric-packable} if it can be asymptotically packed into every sequence of drawings of the complete graph $K_n$. For example, the set of geometric triangles is geometric-packable due to the existence of Steiner Triple Systems. When $G$ is the $4$-cycle (or $4$-cycle with a chord), we show that the set of plane drawings of $G$ is geometric-packable. In contrast, the analogous statement is false when $G$ is nearly any other planar Hamiltonian graph (with at most 3 possible exceptions). A convex geometric graph is {\em convex-packable} if it can be asymptotica…
The k-limited packing number, $L_k(G)$, of a graph $G$, introduced by Gallant, Gunther, Hartnell, and Rall, is the maximum cardinality of a set $X$ of vertices of $G$ such that every vertex of $G$ has at most $k$ elements of $X$ in its closed neighbourhood. The main aim in this paper is to prove the best-possible result that if $G$ is a cubic graph, then $L_2(G) \geq |V (G)|/3$, improving the previous lower bound given by Gallant, \emph{et al.} In addition, we construct an infinite family of graphs to show that lower bounds given by Gagarin and Zverovich are asymptotically best-possible, up to…
Let $G$ and $H$ be graphs. We say that $P$ is an $H$-packing of $G$ if $P$ is a set of edge-disjoint copies of $H$ in $G$. An $H$-packing $P$ is maximal if there is no other $H$-packing of $G$ that properly contains $P$. Packings of maximum cardinality have been studied intensively, with several recent breakthrough results. Here, we consider minimum cardinality maximal packings. An $H$-packing $P$ is clumsy if it is maximal of minimum size. Let $cl(G,H)$ be the size of a clumsy $H$-packing of $G$. We provide nontrivial bounds for $cl(G,H)$, and in many cases asymptotically determine $cl(G,H)$ …
We prove that in every $2$-edge-colouring of $K_n$ there is a collection of $n^2/12 + o(n^2)$ edge-disjoint monochromatic triangles, thus confirming a conjecture of Erdős. We also prove a corresponding stability result, showing that $2$-colourings that are close to attaining the aforementioned bound have a colour class which is close to bipartite. As part of our proof, we confirm a recent conjecture of Tyomkyn about the fractional version of this problem.
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 condition which ensures that a graph G has a perfect H-packing. More precisely, let δ_{\rm Ore} (H,n) be the smallest number k such that every graph G whose order n is divisible by |H| and with d(x)+d(y)\geq k for all non-adjacent x \not = y \in V(G) contains a perfect H-packing. We determine \lim_{n\to \infty} δ_{\rm Ore} (H,n)/n.
We introduce a new procedure for generating the binomial random graph/hypergraph models, referred to as \emph{online sprinkling}. As an illustrative application of this method, we show that for any fixed integer $k\geq 3$, the binomial $k$-uniform random hypergraph $H^{k}_{n,p}$ contains $N:=(1-o(1))\binom{n-1}{k-1}p$ edge-disjoint perfect matchings, provided $p\geq \frac{\log^{C}n}{n^{k-1}}$, where $C:=C(k)$ is an integer depending only on $k$. Our result for $N$ is asymptotically best optimal and for $p$ is optimal up to the $polylog(n)$ factor.
Tuza conjectured that for every graph $G$, the maximum size $ν$ of a set of edge-disjoint triangles and minimum size $τ$ of a set of edges meeting all triangles satisfy $τ\leq 2ν$. We consider an edge-weighted version of this conjecture, which amounts to packing and covering triangles in multigraphs. Several known results about the original problem are shown to be true in this context, and some are improved. In particular, we answer a question of Krivelevich who proved that $τ\leq 2ν^*$ (where $ν^*$ is the fractional version of $ν$), and asked if this is tight. We prove that $τ\leq 2ν^*-\frac{…
It is well known that a graph with $m$ edges can be made triangle-free by removing (slightly less than) $m/2$ edges. On the other hand, there are many classes of graphs which are hard to make triangle-free in the sense that it is necessary to remove roughly $m/2$ edges in order to eliminate all triangles. It is proved that dense graphs that are hard to make triangle-free, have a large packing of pairwise edge-disjoint triangles. In particular, they have more than $m(1/4+cβ^2)$ pairwise edge-disjoint triangles where $β$ is the density of the graph and $c$ is an absolute constant. This improves …
Let $G$ be an undirected network with a distinguished set of terminals $T \subseteq V(G)$ and edge capacities $cap: E(G) \rightarrow \mathbb{R}_+$. By an odd $T$-walk we mean a walk in $G$ (with possible vertex and edge self-intersections) connecting two distinct terminals and consisting of an odd number of edges. Inspired by the work of Schrijver and Seymour on odd path packing for two terminals, we consider packings of odd $T$-walks subject to capacities $cap$. First, we present a strongly polynomial time algorithm for constructing a maximum fractional packing of odd $T$-walks. For even inte…
Let $G$ and $H$ be $k$-graphs ($k$-uniform hypergraphs); then a perfect $H$-packing in $G$ is a collection of vertex-disjoint copies of $H$ in $G$ which together cover every vertex of $G$. For any fixed $H$ let $δ(H, n)$ be the minimum $δ$ such that any $k$-graph $G$ on $n$ vertices with minimum codegree $δ(G) \geq δ$ contains a perfect $H$-packing. The problem of determining $δ(H, n)$ has been widely studied for graphs (i.e. $2$-graphs), but little is known for $k \geq 3$. Here we determine the asymptotic value of $δ(H, n)$ for all complete $k$-partite $k$-graphs $H$, as well as a wide class …
A Hamilton cycle in a digraph is a cycle that passes through all the vertices, where all the arcs are oriented in the same direction. The problem of finding Hamilton cycles in directed graphs is well studied and is known to be hard. One of the main reasons for this, is that there is no general tool for finding Hamilton cycles in directed graphs comparable to the so called Pos?a `rotation-extension' technique for the undirected analogue. Let ${\mathcal D}(n,p)$ denote the random digraph on vertex set $[n]$, obtained by adding each directed edge independently with probability $p$. Here, we prese…
We say that a graph $G$ has a perfect $H$-packing if there exists a set of vertex-disjoint copies of $H$ which cover all the vertices in $G$. The seminal Hajnal--Szemerédi theorem characterises the minimum degree that ensures a graph $G$ contains a perfect $K_r$-packing. Balogh, Kostochka and Treglown proposed a degree sequence version of the Hajnal--Szemerédi theorem which, if true, gives a strengthening of the Hajnal--Szemerédi theorem. In this paper we prove this conjecture asymptotically. Another fundamental result in the area is the Alon--Yuster theorem which gives a minimum degree condit…
An old conjecture of Zs. Tuza says that for any graph $G$, the ratio of the minimum size, $τ_3(G)$, of a set of edges meeting all triangles to the maximum size, $ν_3(G)$, of an edge-disjoint triangle packing is at most 2. Here, disproving a conjecture of R. Yuster, we show that for any fixed, positive $α$ there are arbitrarily large graphs $G$ of positive density satisfying $τ_3(G)>(1-o(1))|G|/2$ and $ν_3(G)<(1+α)|G|/4$.
Conlon, Gowers, Samotij, and Schacht showed that for a given graph $H$ and a constant $γ> 0$, there exists $C > 0$ such that if $p \ge Cn^{-1/m_2(H)}$ then asymptotically almost surely every spanning subgraph $G$ of the random graph $\mathcal{G}(n,p)$ with minimum degree at least $δ(G) \ge (1 - 1/χ_{\mathrm{cr}}(H) + γ)np$ contains an $H$-packing that covers all but at most $γn$ vertices. Here, $χ_{\mathrm{cr}}(H)$ denotes the critical chromatic threshold, a parameter introduced by Komlós. We show that this theorem can be bootstraped to obtain an $H$-packing covering all but at most $γ(C/p)^{m…
Fractional graph isomorphism is the linear relaxation of an integer programming formulation of graph isomorphism. It preserves some invariants of graphs, like degree sequences and equitable partitions, but it does not preserve others like connectivity, clique and independence numbers, chromatic number, vertex and edge cover numbers, matching number, domination and total domination numbers. In this work, we extend the concept of fractional graph isomorphism to hypergraphs, and give an alternative characterization, analogous to one of those that are known for graphs. With this new concept we pro…
A vertex subset M of a graph G is a multipacking if for each vertex v, and each positive integer s less than or equal to the diameter of G, v is within distance s of at most s vertices of M. The multipacking number of a graph is the maximum cardinality of a multipacking of G. A generalization of 2-packings, multipackings offer interesting insight into the minimum cost broadcast domination problem. This paper surveys recent results in the study of multipackings, including the equality of the multipacking number and broadcast number in trees, an extension of Farber's Algorithm for finding domina…
A subset $C$ of edges in a $k$-uniform hypergraph $H$ is a \emph{loose Hamilton cycle} if $C$ covers all the vertices of $H$ and there exists a cyclic ordering of these vertices such that the edges in $C$ are segments of that order and such that every two consecutive edges share exactly one vertex. The binomial random $k$-uniform hypergraph $H^k_{n,p}$ has vertex set $[n]$ and an edge set $E$ obtained by adding each $k$-tuple $e\in \binom{[n]}{k}$ to $E$ with probability $p$, independently at random. Here we consider the problem of finding edge-disjoint loose Hamilton cycles covering all but $…
Let $Γ$ be a countable group and let $G$ be the Schreier graph of the free part of the Bernoulli shift of $Γ$ (with respect to some finite subset $F \subseteq Γ$). We show that the Borel fractional chromatic number of $G$ is equal to $1$ over the measurable independence number of $G$. As a consequence, we asymptotically determine the Borel fractional chromatic number of $G$ when $Γ$ is the free group, answering a question of Meehan.
We define non-commutative versions of the vertex packing polytope, the theta convex body and the fractional vertex packing polytope of a graph, and establish a quantum version of the Sandwich Theorem of Grötschel, Lovász and Schrijver. We define new non-commutative versions of the Lovász number of a graph which lead to an upper bound of the zero-error capacity of the corresponding quantum channel that can be genuinely better than the one established previously by Duan, Severini and Winter. We define non-commutative counterparts of widely used classical graph parameters and establish their inte…
Edge connectivity and vertex connectivity are two fundamental concepts in graph theory. Although by now there is a good understanding of the structure of graphs based on their edge connectivity, our knowledge in the case of vertex connectivity is much more limited. An essential tool in capturing edge connectivity are edge-disjoint spanning trees. The famous results of Tutte and Nash-Williams show that a graph with edge connectivity $λ$ contains $\floor{λ/2}$ edge-disjoint spanning trees. We present connected dominating set (CDS) partition and packing as tools that are analogous to edge-disjoin…
