Let G be a graph with maximum degree Δwhose vertex set is partitioned into parts V(G) = V_1 \cup ... \cup V_r. A transversal is a subset of V(G) containing exactly one vertex from each part V_i. If it is also an independent set, then we call it an independent transversal. The local degree of G is the maximum number of neighbors of a vertex v in a part V_i, taken over all choices of V_i and v \not \in V_i. We prove that for every fixed ε> 0, if all part sizes |V_i| >= (1+ε)Δand the local degree of G is o(Δ), then G has an independent transversal for sufficiently large Δ. This extends several previous results and settles (in a stronger form) a conjecture of Aharoni and Holzman. We then generalize this result to transversals that induce no cliques of size s. (Note that independent transversals correspond to s=2.) In that context, we prove that parts of size |V_i| >= (1+ε)[Δ/(s-1)] and local degree o(Δ) guarantee the existence of such a transversal, and we provide a construction that shows this is asymptotically tight.
In 1994, Erdős, Gyárfás and Łuczak 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 $n\le (1-o(1))k^2$, there wi…
We show that every locally sparse graph contains a linearly sized expanding subgraph. For constants $c_1>c_2>1$, $0<α<1$, a graph $G$ on $n$ vertices is called a $(c_1,c_2,α)$-graph if it has at least $c_1n$ edges, but every vertex subset $W\subset V(G)$ of size $|W|\le αn$ spans less than $c_2|W|$ edges. We prove that every $(c_1,c_2,α)$-graph with bounded degrees contains an induced expander on linearly many vertices. The proof can be made algorithmic. We then discuss several applications of our main result to random graphs, to problems about embedding graph minors, and to positional games.
A famous theorem of Kirkman says that there exists a Steiner triple system of order $n$ if and only if $n\equiv 1,3\mod{6}$. In 1973, Erdős conjectured that one can find so-called `sparse' Steiner triple systems. Roughly speaking, the aim is to have at most $j-3$ triples on every set of $j$ points, which would be best possible. (Triple systems with this sparseness property are also referred to as having high girth.) We prove this conjecture asymptotically by analysing a natural generalization of the triangle removal process. Our result also solves a problem posed by Lefmann, Phelps and Rödl as…
Supersaturation of induced even cycles in locally sparse graphs
Charles University, Faculty of Mathematics and Physics（查理大学数学物理学院）；EPFL（洛桑联邦理工学院）；University of Oxford（牛津大学）；École Polytechnique, Institut Polytechnique de Paris（巴黎综合理工学院，巴黎理工学院）
An $(n,s,q)$-graph is an $n$-vertex multigraph where every set of $s$ vertices spans at most $q$ edges. In this paper, we determine the maximum product of the edge multiplicities in $(n,s,q)$-graphs if the congruence class of $q$ modulo ${s\choose 2}$ is in a certain interval of length about $3s/2$. The smallest case that falls outside this range is $(s,q)=(4,15)$, and here the answer is $a^{n^2+o(n^2)}$ where $a$ is transcendental assuming Schanuel's conjecture. This could indicate the difficulty of solving the problem in full generality. Many of our results can be seen as extending work by B…
We give a short proof of a bound on the list chromatic number of graphs $G$ of maximum degree $Δ$ where each neighbourhood has density at most $d$, namely $χ_\ell(G) \le (1+o(1)) \fracΔ{\ln \fracΔ{d+1}}$ as $\fracΔ{d+1} \to \infty$. This bound is tight up to an asymptotic factor $2$, which is the best possible barring a breakthrough in Ramsey theory, and strengthens results due to Vu, and more recently Davies, P., Kang, and Sereni. Our proof relies on the first moment method, and adapts a clever counting argument developed by Rosenfeld in the context of non-repetitive colourings. As a final to…
Developing further Stein's recent notion of relative end degrees in infinite graphs, we investigate which degree assumptions can force a locally finite graph to contain a given finite minor, or a finite subgraph of given minimum degree. This is part of a wider project which seeks to develop an extremal theory of sparse infinite graphs.
We provide a gentle introduction, aimed at non-experts, to Borel combinatorics that studies definable graphs on topological spaces. This is an emerging field on the borderline between combinatorics and descriptive set theory with deep connections to many other areas. After giving some background material, we present in careful detail some basic tools and results on the existence of Borel satisfying assignments: Borel versions of greedy algorithms and augmenting procedures, local rules, Borel transversals, etc. Also, we present the construction of Andrew Marks of acyclic Borel graphs for which …
We develop an algorithmic framework for graph colouring that reduces the problem to verifying a local probabilistic property of the independent sets. With this we give, for any fixed $k\ge 3$ and $\varepsilon>0$, a randomised polynomial-time algorithm for colouring graphs of maximum degree $Δ$ in which each vertex is contained in at most $t$ copies of a cycle of length $k$, where $1/2\le t\le Δ^\frac{2\varepsilon}{1+2\varepsilon}/(\logΔ)^2$, with $\lfloor(1+\varepsilon)Δ/\log(Δ/\sqrt t)\rfloor$ colours. This generalises and improves upon several notable results including those of Kim (1995) an…
We prove several results on approximate decompositions of edge-coloured quasirandom graphs into rainbow spanning structures. More precisely, we say that an edge-colouring of a graph is locally $\ell$-bounded if no vertex is incident to more than $\ell$ edges of any given colour, and that it is (globally) $g$-bounded if no colour appears more than $g$ times in the colouring. Note that every proper colouring of an $n$-vertex graph is locally $1$-bounded, and (globally) $n/2$-bounded. Our results imply the following: (i) The existence of approximate decompositions of edge-coloured $K_n$ into rain…
In this paper, we study a primal and dual relationship about triangles: For any graph $G$, let $ν(G)$ be the maximum number of edge-disjoint triangles in $G$, and $τ(G)$ be the minimum subset $F$ of edges such that $G \setminus F$ is triangle-free. It is easy to see that $ν(G) \leq τ(G) \leq 3 ν(G)$, and in fact, this rather obvious inequality holds for a much more general primal-dual relation between $k$-hyper matching and covering in hypergraphs. Tuza conjectured in $1981$ that $τ(G) \leq 2 ν(G)$, and this question has received attention from various groups of researchers in discrete mathema…
Fast Shortest Path in Graphs With Sparse Signed Tree Models and Applications
CNRS, ENS de Lyon, Université Claude Bernard Lyon 1（法国国家科学研究中心、里昂高等师范学院、里昂第一大学）；Discrete Mathematics Group, Institute for Basic Science (IBS)（基础科学研究院离散数学组）；School of Computing, KAIST（韩国科学技术院计算机学院）
We use an inequality of Sidorenko to show a general relation between local and global subgraph counts and degree moments for locally weakly convergent sequences of sparse random graphs. This yields an optimal criterion to check when the asymptotic behaviour of graph statistics such as the clustering coefficient and assortativity is determined by the local weak limit. As an application we obtain new facts for several common models of sparse random intersection graphs where the local weak limit, as we see here, is a simple random clique tree corresponding to a certain two-type Galton-Watson bran…
