Asymptotically optimal KkK_k-packings of dense graphs via fractional KkK_k-decompositions

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.Comment: 12 page

Computing the diameter polynomially faster than APSP

We present a new randomized algorithm for computing the diameter of a weighted directed graph. The algorithm runs in \Ot(M^{\w/(\w+1)}n^{(\w^2+3)/(\w+1)}) time, where \w < 2.376 is the exponent of fast matrix multiplication, $n$ is the number of vertices of the graph, and the edge weights are integers in $\{-M,...,0,...,M\}$. For bounded integer weights the running time is $O(n^{2.561})$ and if \w=2+o(1) it is \Ot(n^{7/3}). This is the first algorithm that computes the diameter of an integer weighted directed graph polynomially faster than any known All-Pairs Shortest Paths (APSP) algorithm. For bounded integer weights, the fastest algorithm for APSP runs in $O(n^{2.575})$ time for the present value of \w and runs in \Ot(n^{2.5}) time if \w=2+o(1). For directed graphs with {\em positive} integer weights in $\{1,...,M\}$ we obtain a deterministic algorithm that computes the diameter in \Ot(Mn^\w) time. This extends a simple \Ot(n^\w) algorithm for computing the diameter of an {\em unweighted} directed graph to the positive integer weighted setting and is the first algorithm in this setting whose time complexity matches that of the fastest known Diameter algorithm for {\em undirected} graphs. The diameter algorithms are consequences of a more general result. We construct algorithms that for any given integer $d$, report all ordered pairs of vertices having distance {\em at most} $d$. The diameter can therefore be computed using binary search for the smallest $d$ for which all pairs are reported.Comment: revised to handle negative weights; faster algorithm for positive weights; added observation regarding the unweighted cas

Families of trees decompose the random graph in any arbitrary way

Let $F=\{H_1,...,H_k\}$ be a family of graphs. A graph $G$ with $m$ edges is called {\em totally $F$-decomposable} if for {\em every} linear combination of the form $\alpha_1 e(H_1) + ... + \alpha_k e(H_k) = m$ where each $\alpha_i$ is a nonnegative integer, there is a coloring of the edges of $G$ with $\alpha_1+...+\alpha_k$ colors such that exactly $\alpha_i$ color classes induce each a copy of $H_i$, for $i=1,...,k$. We prove that if $F$ is any fixed family of trees then $\log n/n$ is a sharp threshold function for the property that the random graph $G(n,p)$ is totally $F$-decomposable. In particular, if $H$ is a tree, then $\log n/n$ is a sharp threshold function for the property that $G(n,p)$ contains $\lfloor e(G)/e(H) \rfloor$ edge-disjoint copies of $H$.Comment: 20 page

Vector clique decompositions

Let $F_k$ be the set of graphs on $k$ vertices. For a graph $G$, a $k$-decomposition is a set of induced subgraphs of $G$, each isomorphic to an element of $F_k$, such that each pair of vertices of $G$ is in exactly one element of the set. A fundamental result of Wilson is that for all $n=|V(G)|$ sufficiently large, $G$ has a $k$-decomposition if and only if $G$ is $k$-divisible. Let ${\bf v} \in {\mathbb R}^{|F_k|}$ be indexed by $F_k$. For a $k$-decomposition $L$ of $G$, let $\nu_{\bf v}(L) = \sum_{F \in F_k} {\bf v}_F d_{L,F}$ where $d_{L,F}$ is the fraction of elements of $L$ isomorphic to $F$. Let $\nu_{\bf v}(G) = \max_{L} \nu_{\bf v}(L)$ and $\nu_{\bf v}(n)=\min\{\nu_{\bf v}(G):|V(G)|=n\}$. It is not difficult to prove that the sequence $\nu_{\bf v}(n)$ has a limit so let $\nu_{\bf v} = \lim_{n \rightarrow \infty} \nu_{\bf v}(n)$. Replacing $k$-decompositions with their fractional relaxations, one obtains the (polynomial time computable) fractional analogue $\nu_{\bf v}^*(G)$ and corresponding fractional values $\nu^*_{\bf v}(n)$ and $\nu^*_{\bf v}$. Our first main result is that for each ${\bf v} \in {\mathbb R}^{|F_k|}$ $$\nu_{\bf v} = \nu^*_{\bf v}\;.$$ Further, there is a polynomial time algorithm that produces a decomposition $L$ of a $k$-decomposable graph such that $\nu_{\bf v}(L) \ge \nu_{\bf v} - o_n(1)$. A similar result holds when $F_k$ is the family of all tournaments on $k$ vertices and when $F_k$ is the family of all edge-colorings of $K_k$. We use these results to obtain new and improved bounds on several decomposition results. For example, we prove that every $n$-vertex tournament which is $3$-divisible has a triangle decomposition in which the number of directed triangles is less than $0.0222n^2(1+o(1))$ and that every $5$-decomposable $n$-vertex graph has a $5$-decomposition in which the fraction of cycles of length $5$ is $o_n(1)$.Comment: 30 page

Equitable coloring of k-uniform hypergraphs

Let $H$ be a $k$-uniform hypergraph with $n$ vertices. A {\em strong $r$-coloring} is a partition of the vertices into $r$ parts, such that each edge of $H$ intersects each part. A strong $r$-coloring is called {\em equitable} if the size of each part is $\lceil n/r \rceil$ or $\lfloor n/r \rfloor$. We prove that for all $a \geq 1$, if the maximum degree of $H$ satisfies $\Delta(H) \leq k^a$ then $H$ has an equitable coloring with $\frac{k}{a \ln k}(1-o_k(1))$ parts. In particular, every $k$-uniform hypergraph with maximum degree $O(k)$ has an equitable coloring with $\frac{k}{\ln k}(1-o_k(1))$ parts. The result is asymptotically tight. The proof uses a double application of the non-symmetric version of the Lov\'asz Local Lemma.Comment: 10 Page

On the exact maximum induced density of almost all graphs and their inducibility

Let $H$ be a graph on $h$ vertices. The number of induced copies of $H$ in a graph $G$ is denoted by $i_H(G)$. Let $i_H(n)$ denote the maximum of $i_H(G)$ taken over all graphs $G$ with $n$ vertices. Let $f(n,h) = \Pi_{i}^h a_i$ where $\sum_{i=1}^h a_i = n$ and the $a_i$ are as equal as possible. Let $g(n,h) = f(n,h) + \sum_{i=1}^h g(a_i,h)$. It is proved that for almost all graphs $H$ on $h$ vertices it holds that $i_H(n)=g(n,h)$ for all $n \le 2^{\sqrt{h}}$. More precisely, we define an explicit graph property ${\cal P}_h$ which, when satisfied by $H$, guarantees that $i_H(n)=g(n,h)$ for all $n \le 2^{\sqrt{h}}$. It is proved, in particular, that a random graph on $h$ vertices satisfies ${\cal P}_h$ with probability $1-o_h(1)$. Furthermore, all extremal $n$-vertex graphs yielding $i_H(n)$ in the aforementioned range are determined. We also prove a stability result. For $H \in {\cal P}_h$ and a graph $G$ with $n \le 2^{\sqrt{h}}$ vertices satisfying $i_H(G) \ge f(n,h)$, it must be that $G$ is obtained from a balanced blowup of $H$ by adding some edges inside the blowup parts. The {\em inducibility} of $H$ is $i_H = \lim_{n \rightarrow \infty} i_H(n)/\binom{n}{h}$. It is known that $i_H \ge h!/(h^h-h)$ for all graphs $H$ and that a random graph $H$ satisfies almost surely that $i_H \le h^{3\log h}h!/(h^h-h)$. We improve upon this upper bound almost matching the lower bound. It is shown that a graph $H$ which satisfies ${\cal P}_h$ has $i_H =(1+O(h^{-h^{1/3}}))h!/(h^h-h)$.Comment: 27 page

Integer and fractional packing of families of graphs

Let ${\cal F}$ be a family of graphs. For a graph $G$, the {\em ${\cal F}$-packing number}, denoted $\nu_{{\cal F}}(G)$, is the maximum number of pairwise edge-disjoint elements of ${\cal F}$ in $G$. A function $\psi$ from the set of elements of ${\cal F}$ in $G$ to $[0,1]$ is a {\em fractional ${\cal F}$-packing} of $G$ if $\sum_{e \in H \in {\cal F}} {\psi(H)} \leq 1$ for each $e \in E(G)$. The {\em fractional ${\cal F}$-packing number}, denoted $\nu^*_{{\cal F}}(G)$, is defined to be the maximum value of $\sum_{H \in {{G} \choose {{\cal F}}}} \psi(H)$ over all fractional ${\cal F}$-packings $\psi$. Our main result is that $\nu^*_{{\cal F}}(G)-\nu_{{\cal F}}(G) = o(|V(G)|^2)$. Furthermore, a set of $\nu_{{\cal F}}(G) -o(|V(G)|^2)$ edge-disjoint elements of ${\cal F}$ in $G$ can be found in randomized polynomial time. For the special case ${\cal F}=\{H_0\}$ we obtain a significantly simpler proof of a recent difficult result of Haxell and R\"odl \cite{HaRo} that $\nu^*_{H_0}(G)-\nu_{H_0}(G) = o(|V(G)|^2)$.Comment: 8 page

Quasi-randomness is determined by the distribution of copies of a fixed graph in equicardinal large sets

For every fixed graph $H$ and every fixed $0 < \alpha < 1$, we show that if a graph $G$ has the property that all subsets of size $\alpha n$ contain the ``correct'' number of copies of $H$ one would expect to find in the random graph $G(n,p)$ then $G$ behaves like the random graph $G(n,p)$; that is, it is $p$-quasi-random in the sense of Chung, Graham, and Wilson. This solves a conjecture raised by Shapira and solves in a strong sense an open problem of Simonovits and S\'os.Comment: 7 page

Mean Ramsey-Tur\'an numbers

A $\rho$-mean coloring of a graph is a coloring of the edges such that the average number of colors incident with each vertex is at most $\rho$. For a graph $H$ and for $\rho \geq 1$, the {\em mean Ramsey-Tur\'an number} $RT(n,H,\rho-mean)$ is the maximum number of edges a $\rho$-mean colored graph with $n$ vertices can have under the condition it does not have a monochromatic copy of $H$. It is conjectured that $RT(n,K_m,2-mean)=RT(n,K_m,2)$ where $RT(n,H,k)$ is the maximum number of edges a $k$ edge-colored graph with $n$ vertices can have under the condition it does not have a monochromatic copy of $H$. We prove the conjecture holds for $K_3$. We also prove that $RT(n,H,\rho-mean) \leq RT(n,K_{\chi(H)},\rho-mean)+o(n^2)$. This result is tight for graphs $H$ whose clique number equals their chromatic number. In particular we get that if $H$ is a 3-chromatic graph having a triangle then $RT(n,H,2-mean) = RT(n,K_3,2-mean)+o(n^2)=RT(n,K_3,2)+o(n^2)=0.4n^2(1+o(1))$.Comment: 9 page

Dense graphs with a large triangle cover have a large triangle packing

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 $m(1/4+c\beta^2)$ pairwise edge-disjoint triangles where $\beta$ is the density of the graph and $c$ is an absolute constant. This improves upon a previous $m(1/4-o(1))$ bound which follows from the asymptotic validity of Tuza's conjecture for dense graphs. It is conjectured that such graphs have an asymptotically optimal triangle packing of size $m(1/3-o(1))$. The result is extended to larger cliques and odd cycles