Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userAn average degree condition for independent transversals
68
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
In 1994, ErdH{o}s, Gy{a}rf{a}s and {L}uczak posed the following problem: given disjoint vertex sets $V_1,dots,V_n$ of size~$k$, with exactly one edge between any pair $V_i,V_j$, how large can $n$ be such that there will always be an independent transversal? They showed that the maximal $n$ is at most $(1+o(1))k^2$, by providing an explicit construction with these parameters and no independent transversal. They also proved a lower bound which is smaller by a $2e$-factor. In this paper, we solve this problem by showing that their upper bound construction is best possible: if $nle (1-o(1))k^2$, there will always be an independent transversal. In fact, this result is a very special case of a much more general theorem which concerns independent transversals in arbitrary partite graphs that are `locally sparse, meaning that the maximum degree between each pair of parts is relatively small. In this setting, Loh and Sudakov provided a global emph{maximum} degree condition for the existence of an independent transversal. We show that this can be relaxed to an emph{average} degree condition. We can also use our new theorem to establish tight bounds for a more general version of the ErdH{o}s--Gy{a}rf{a}s--{L}uczak problem and solve a conjecture of Yuster from 1997. This exploits a connection to the Turan numbers of complete bipartite graphs, which might be of independent interest. Finally, we discuss some partial results on the hypergraph variant of the problem, and formulate a conjecture in this case.
rate research
Read More
An independent transversal of a graph $G$ with a vertex partition $mathcal P$ is an independent set of $G$ intersecting each block of $mathcal P$ in a single vertex. Wanless and Wood proved that if each block of $mathcal P$ has size at least $t$ and the average degree of vertices in each block is at most $t/4$, then an independent transversal of $mathcal P$ exists. We present a construction showing that this result is optimal: for any $varepsilon > 0$ and sufficiently large $t$, there is a family of forests with vertex partitions whose block size is at least $t$, average degree of vertices in each block is at most $(frac14+varepsilon)t$, and there is no independent transversal. This unexpectedly shows that methods related to entropy compression such as the Rosenfeld-Wanles-Wood scheme or the Local Cut Lemma are tight for this problem. Further constructions are given for variants of the problem, including the hypergraph version.
Motivated by a longstanding conjecture of Thomassen, we study how large the average degree of a graph needs to be to imply that it contains a $C_4$-free subgraph with average degree at least $t$. Kuhn and Osthus showed that an average degree bound which is double exponential in t is sufficient. We give a short proof of this bound, before reducing it to a single exponential. That is, we show that any graph $G$ with average degree at least $2^{ct^2log t}$ (for some constant $c>0$) contains a $C_4$-free subgraph with average degree at least $t$. Finally, we give a construction which improves the lower bound for this problem, showing that this initial average degree must be at least $t^{3-o(1)}$.
We consider numbers and sizes of independent sets in graphs with minimum degree at least $d$, when the number $n$ of vertices is large. In particular we investigate which of these graphs yield the maximum numbers of independent sets of different sizes, and which yield the largest random independent sets. We establish a strengthened form of a conjecture of Galvin concerning the first of these topics.
Given a graph $G$, let $f_{G}(n,m)$ be the minimal number $k$ such that every $k$ independent $n$-sets in $G$ have a rainbow $m$-set. Let $mathcal{D}(2)$ be the family of all graphs with maximum degree at most two. Aharoni et al. (2019) conjectured that (i) $f_G(n,n-1)=n-1$ for all graphs $Ginmathcal{D}(2)$ and (ii) $f_{C_t}(n,n)=n$ for $tge 2n+1$. Lv and Lu (2020) showed that the conjecture (ii) holds when $t=2n+1$. In this article, we show that the conjecture (ii) holds for $tgefrac{1}{3}n^2+frac{44}{9}n$. Let $C_t$ be a cycle of length $t$ with vertices being arranged in a clockwise order. An ordered set $I=(a_1,a_2,ldots,a_n)$ on $C_t$ is called a $2$-jump independent $n$-set of $C_t$ if $a_{i+1}-a_i=2pmod{t}$ for any $1le ile n-1$. We also show that a collection of 2-jump independent $n$-sets $mathcal{F}$ of $C_t$ with $|mathcal{F}|=n$ admits a rainbow independent $n$-set, i.e. (ii) holds if we restrict $mathcal{F}$ on the family of 2-jump independent $n$-sets. Moreover, we prove that if the conjecture (ii) holds, then (i) holds for all graphs $Ginmathcal{D}(2)$ with $c_e(G)le 4$, where $c_e(G)$ is the number of components of $G$ isomorphic to cycles of even lengths.
ErdH{o}s posed the problem of finding conditions on a graph $G$ that imply the largest number of edges in a triangle-free subgraph is equal to the largest number of edges in a bipartite subgraph. We generalize this problem to general cases. Let $delta_r$ be the least number so that any graph $G$ on $n$ vertices with minimum degree $delta_rn$ has the property $P_{r-1}(G)=K_rf(G),$ where $P_{r-1}(G)$ is the largest number of edges in an $(r-1)$-partite subgraph and $K_rf(G)$ is the largest number of edges in a $K_r$-free subgraph. We show that $frac{3r-4}{3r-1}<delta_rlefrac{4(3r-7)(r-1)+1}{4(r-2)(3r-4)}$ when $rge4.$ In particular, $delta_4le 0.9415.$
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
