arXiv subjects

Donglei Yang

Publications and source records attributed to Donglei Yang.

36 records · Page 2Linked to original sources

arXiv2023

Clique-factors in graphs with sublinear $\ell$-independence number

Given a graph $G$ and an integer $\ell\ge 2$, we denote by $α_{\ell}(G)$ the maximum size of a $K_{\ell}$-free subset of vertices in $V(G)$. A recent question of Nenadov and Pehova asks for determining the best possible minimum degree conditions forcing clique-factors in $n$-vertex graphs $G$ with $α_{\ell}(G) = o(n)$, which can be seen as a Ramsey--Turán variant of the celebrated Hajnal--Szemerédi theorem. In this paper we find the asymptotical sharp minimum degree threshold for $K_r$-factors in $n$-vertex graphs $G$ with $α_\ell(G)=n^{1-o(1)}$ for all $r\ge \ell\ge 2$.

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arXiv2023

Balanced subdivisions of cliques in graphs

Given a graph $H$, a balanced subdivision of $H$ is a graph obtained from $H$ by subdividing every edge the same number of times. In 1984, Thomassen conjectured that for each integer $k\ge 1$, high average degree is sufficient to guarantee a balanced subdivision of $K_k$. Recently, Liu and Montgomery resolved this conjecture. We give an optimal estimate up to an absolute constant factor by showing that there exists $c>0$ such that for sufficiently large $d$, every graph with average degree at least $d$ contains a balanced subdivision of a clique with at least $cd^{1/2}$ vertices. It also confirms a conjecture from Verstra{ë}te: every graph of average degree $cd^2$, for some absolute constant $c>0$, contains a pair of disjoint isomorphic subdivisions of the complete graph $K_d$. We also prove that there exists some absolute $c>0$ such that for sufficiently large $d$, every $C_4$-free graph with average degree at least $d$ contains a balanced subdivision of the complete graph $K_{cd}$, which extends a result of Balogh, Liu and Sharifzadeh.

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arXiv2023

Spanning trees in sparse expanders

Given integers $n\ge Δ\ge 2$, let $\mathcal{T}(n, Δ)$ be the collection of all $n$-vertex trees with maximum degree at most $Δ$. A question of Alon, Krivelevich and Sudakov in 2007 asks for determining the best possible spectral gap condition forcing an $(n, d,λ)$-graph to be $\mathcal{T}(n, Δ)$-universal, namely, it contains all members of $\mathcal{T}(n, Δ)$ as a subgraph simultaneously. In this paper we show that for sufficiently large integer $n$ and all $Δ\in \mathbb{N}$, every $(n, d,λ)$-graph with \[ λ\le\frac{d}{2Δ^{5\sqrt{\log n}}} \] is $\mathcal{T}(n, Δ)$-universal. As an immediate corollary, this implies that Alon's ingenious construction of triangle-free sparse expander is $\mathcal{T}(n, Δ)$-universal, which provides an explicit construction of such graphs and thus solves a question of Johannsen, Krivelevich and Samotij. Our main result is formulated under a much more general context, namely, the $(n,d)$-expanders. More precisely, we show that there exist absolute constants $C,c>0$ such that the following statement holds for sufficiently large integer $n$. (1).For all $Δ\in \mathbb{N}$, every $(n, Δ^{5\sqrt{\log n}})$-expander is $\mathcal{T}(n, Δ)$-universal. (2).For all $Δ\in \mathbb{N}$ with $Δ\le c\sqrt{n}$, every $(n, CΔn^{1/2})$-expander is $\mathcal{T}(n, Δ)$-universal. Both results significantly improve a result of Johannsen, Krivelevich and Samotij, and have further implications in locally sparse expanders and Maker-Breaker games that also improve previously known results drastically.

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arXiv2023

Compatible Powers of Hamilton Cycles in Dense Graphs

Motivated by the concept of transition system investigated by Kotzig in 1968, Krivelevich, Lee and Sudakov proposed a more general notion of incompatibility system to formulate the robustness of Hamiltonicity of Dirac graphs. Given a graph $G=(V,E)$, an {\em incompatibility system} $\mathcal{F}$ over $G$ is a family $\mathcal{F}=\{F_v\}_{v\in V}$ such that for every $v\in V$, $F_v$ is a family of edge pairs in $\{\{e,e'\}: e\ne e'\in E, e\cap e'=\{v\}\}$. An incompatibility system $\mathcal{F}$ is \emph{$Δ$-bounded} if for every vertex $v$ and every edge $e$ incident with $v$, there are at most $Δ$ pairs in $F_v$ containing $e$. A subgraph $H$ of $G$ is \emph{compatible} (with respect to $\mathcal{F}$) if every pair of adjacent edges $e,e'$ of $H$ satisfies $\{e,e'\} \notin F_v$, where $v=e\cap e'$. Krivelevich, Lee and Sudakov proved that there is an universal constant $μ>0$ such that for every $μn$-bounded incompatibility system $\mathcal{F}$ over a Dirac graph, there exists a compatible Hamilton cycle, which resolves a conjecture of Häggkvist from 1988. We study high powers of Hamilton cycles in this context and show that for every $γ>0$ and $k\in\mathbb{N}$, there exists a constant $μ>0$ such that for sufficiently large $n\in\mathbb{N}$ and every $μn$-bounded incompatibility system over an $n$-vertex graph $G$ with $δ(G)\ge(\frac{k}{k+1}+γ)n$, there exists a compatible $k$-th power of a Hamilton cycle in $G$. Moreover, we give a construction which has minimum degree $\frac{k}{k+1}n+Ω(n)$ and contains no compatible $k$-th power of a Hamilton cycle.

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arXiv2023

Spanning trees in graphs without large bipartite holes

We show that for any $\varepsilon>0$ and $Δ\in\mathbb{N}$, there exists $α>0$ such that for sufficiently large $n$, every $n$-vertex graph $G$ satisfying that $δ(G)\geq\varepsilon n$ and $e(X, Y)>0$ for every pair of disjoint vertex sets $X, Y\subseteq V(G)$ of size $αn$ contains all spanning trees with maximum degree at most $Δ$. This strengthens a result of Böttcher et al.

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arXiv2022

Shape of the asymptotic maximum sum-free sets in integer lattice grids

We determine the shape of all sum-free sets in $\{1,2,\ldots,n\}^2$ of size close to the maximum $\frac{3}{5}n^2$, solving a problem of Elsholtz and Rackham. We show that all such asymptotic maximum sum-free sets lie completely in the stripe $\frac{4}{5}n-o(n)\le x+y\le\frac{8}{5}n+ o(n)$. We also determine for any positive integer $p$ the maximum size of a subset $A\subseteq \{1,2,\ldots,n\}^2$ which forbids the triple $(x,y,z)$ satisfying $px+py=z$.

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arXiv2022

Clique immersion in graphs without fixed bipartite graph

A graph $G$ contains $H$ as an \emph{immersion} if there is an injective mapping $ϕ: V(H)\rightarrow V(G)$ such that for each edge $uv\in E(H)$, there is a path $P_{uv}$ in $G$ joining vertices $ϕ(u)$ and $ϕ(v)$, and all the paths $P_{uv}$, $uv\in E(H)$, are pairwise edge-disjoint. An analogue of Hadwiger's conjecture for the clique immersions by Lescure and Meyniel, and independently by Abu-Khzam and Langston, states that every graph $G$ contains $K_{χ(G)}$ as an immersion. We prove that for any constant $\varepsilon>0$ and integers $s,t\ge2$, there exists $d_0=d_0(\varepsilon,s,t)$ such that every $K_{s,t}$-free graph $G$ with $d(G)\ge d_0$ contains a clique immersion of order $(1-\varepsilon)d(G)$. This implies that the above-mentioned conjecture is asymptotically true for graphs without a fixed complete bipartite graph.

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arXiv2022

$H$-factors in graphs with small independence number

Let $H$ be an $h$-vertex graph. The vertex arboricity $ar(H)$ of $H$ is the least integer $r$ such that $V(H)$ can be partitioned into $r$ parts and each part induces a forest in $H$. We show that for sufficiently large $n\in h\mathbb{N}$, every $n$-vertex graph $G$ with $δ(G)\geq \max\left\{\left(1-\frac{2}{f(H)}+o(1)\right)n, \left(\frac{1}{2}+o(1)\right)n\right\}$ and $α(G)=o(n)$ contains an $H$-factor, where $f(H)=2ar(H)$ or $2ar(H)-1$. The result can be viewed an analogue of the Alon--Yuster theorem \cite{MR1376050} in Ramsey--Turán theory, which generalises the results of Balogh--Molla--Sharifzadeh~\cite{MR3570984} and Knierm--Su~\cite{MR4193066} on clique factors. In particular the degree conditions are asymptotically sharp for infinitely many graphs $H$ which are not cliques.

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arXiv2021

Embedding clique-factors in graphs with low $\ell$-independence number

The following question was proposed by Nenadov and Pehova and reiterated by Knierim and Su: Given integers $\ell,r$ and $n$ with $n\in r\mathbb{N}$, is it true that every $n$-vertex graph $G$ with $δ(G) \ge \max \{ \frac{1}{2},\frac{r - \ell}{r} \}n + o(n)$ and $α_{\ell}(G) = o(n) $ contains a $K_{r}$-factor? We give a negative answer for the case when $\ell\ge \frac{3r}{4}$ by giving a family of constructions using the so-called cover thresholds and show that the minimum degree condition given by our construction is asymptotically best possible. That is, for all integers $r,\ell$ with $r > \ell \ge \frac{3}{4}r$ and $μ>0$, there exist $α> 0$ and $N$ such that for every $n\in r\mathbb{N}$ with $n>N$, every $n$-vertex graph $G$ with $δ(G) \ge \left( \frac{1}{2-\varrho_{\ell}(r-1)} + μ\right)n $ and $α_{\ell}(G) \le αn$ contains a $K_{r}$-factor. Here $\varrho_{\ell}(r-1)$ is the Ramsey--Turán density for $K_{r-1}$ under the $\ell$-independence number condition.

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arXiv2021

Properly colored short cycles in edge-colored graphs

Properly colored cycles in edge-colored graphs are closely related to directed cycles in oriented graphs. As an analogy of the well-known Caccetta-Häggkvist Conjecture, we study the existence of properly colored cycles of bounded length in an edge-colored graph. We first prove that for all integers $s$ and $t$ with $t\geq s\geq2$, every edge-colored graph $G$ with no properly colored $K_{s,t}$ contains a spanning subgraph $H$ which admits an orientation $D$ such that every directed cycle in $D$ is a properly colored cycle in $G$. Using this result, we show that for $r\geq4$, if the Caccetta-Häggkvist Conjecture holds , then every edge-colored graph of order $n$ with minimum color degree at least $n/r+2\sqrt{n}+1$ contains a properly colored cycle of length at most $r$. In addition, we also obtain an asymptotically tight total color degree condition which ensures a properly colored (or rainbow) $K_{s,t}$.

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arXiv2020

Flexibility of planar graphs without $C_4$ and $C_5$

Let $G$ be a $\{C_4, C_5\}$-free planar graph with a list assignment $L$. Suppose a preferred color is given for some of the vertices. We prove that if all lists have size at least four, then there exists an $L$-coloring respecting at least a constant fraction of the preferences.

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arXiv2019

Antimagic orientations of graphs with given independence number

Given a digraph $D$ with $m$ arcs and a bijection $τ: A(D)\rightarrow \{1, 2, \ldots, m\}$, we say $(D, τ)$ is an antimagic orientation of a graph $G$ if $D$ is an orientation of $G$ and no two vertices in $D$ have the same vertex-sum under $τ$, where the vertex-sum of a vertex $u$ in $D$ under $τ$ is the sum of labels of all arcs entering $u$ minus the sum of labels of all arcs leaving $u$. Hefetz, Mütze, and Schwartz in 2010 initiated the study of antimagic orientations of graphs, and conjectured that every connected graph admits an antimagic orientation. This conjecture seems hard, and few related results are known. However, it has been verified to be true for regular graphs, biregular bipartite graphs, and graphs with large maximum degree. In this paper, we establish more evidence for the aforementioned conjecture by studying antimagic orientations of graphs $G$ with independence number at least $|V(G)|/2$ or at most four. We obtain several results. The method we develop in this paper may shed some light on attacking the aforementioned conjecture.

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arXiv2019

Antimagic orientations of graphs with large maximum degree

Given a digraph $D$ with $m $ arcs, a bijection $τ: A(D)\rightarrow \{1, 2, \ldots, m\}$ is an antimagic labeling of $D$ if no two vertices in $D$ have the same vertex-sum, where the vertex-sum of a vertex $u $ in $D$ under $τ$ is the sum of labels of all arcs entering $u$ minus the sum of labels of all arcs leaving $u$. We say $(D, τ)$ is an antimagic orientation of a graph $G$ if $D$ is an orientation of $G$ and $τ$ is an antimagic labeling of $D$. Motivated by the conjecture of Hartsfield and Ringel from 1990 on antimagic labelings of graphs, Hefetz, Mütze, and Schwartz in 2010 initiated the study of antimagic orientations of graphs, and conjectured that every connected graph admits an antimagic orientation. This conjecture seems hard, and few related results are known. However, it has been verified to be true for regular graphs and biregular bipartite graphs. In this paper, we prove that every connected graph $G$ on $n\ge9$ vertices with maximum degree at least $n-5$ admits an antimagic orientation.

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arXiv2019

A note on antimagic orientations of even regular graphs

Motivated by the conjecture of Hartsfield and Ringel on antimagic labelings of undirected graphs, Hefetz, Mütze, and Schwartz initiated the study of antimagic labelings of digraphs in 2010. Very recently, it has been conjectured in [Antimagic orientation of even regular graphs, J. Graph Theory, 90 (2019), 46-53.] that every graph admits an antimagtic orientation, which is a strengthening of an earlier conjecture of Hefetz, Mütze and Schwartz. In this paper, we prove that every $2d$-regular graph (not necessarily connected) admits an antimagic orientation, where $d\ge2$. Together with known results, our main result implies that the above-mentioned conjecture is true for all regular graphs.

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arXiv2018

Strong list-chromatic index of subcubic graphs

A strong $k$-edge-coloring of a graph G is an edge-coloring with $k$ colors in which every color class is an induced matching. The strong chromatic index of $G$, denoted by $χ'_{s}(G)$, is the minimum $k$ for which $G$ has a strong $k$-edge-coloring. In 1985, Erdős and Nešetřil conjectured that $χ'_{s}(G)\leq\frac{5}{4}Δ(G)^2$, where $Δ(G)$ is the maximum degree of $G$. When $G$ is a graph with maximum degree at most 3, the conjecture was verified independently by Andersen and Horák, Qing, and Trotter. In this paper, we consider the list version of strong edge-coloring. In particular, we show that every subcubic graph has strong list-chromatic index at most 11 and every planar subcubic graph has strong list-chromatic index at most 10.

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arXiv2018

On splitting digraphs

In 1995, Stiebitz asked the following question: For any positive integers $s,t$, is there a finite integer $f(s,t)$ such that every digraph $D$ with minimum out-degree at least $f(s,t)$ admits a bipartition $(A, B)$ such that $A$ induces a subdigraph with minimum out-degree at least $s$ and $B$ induces a subdigraph with minimum out-degree at least $t$? We give an affirmative answer for tournaments, multipartite tournaments, and digraphs with bounded maximum in-degrees. In particular, we show that for every $ε$ with $0<ε<1/2$, there exists an integer $δ_0$ such that every tournament with minimum out-degree at least $δ_0$ admits a bisection $(A, B)$, so that each vertex has at least $(1/2-ε)$ of its out-neighbors in $A$, and in $B$ as well.

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arXiv2018

List strong edge-coloring of graphs with maximum degree 4

A strong edge-coloring of a graph $G$ is an edge-coloring such that any two edges on a path of length three receive distinct colors. We denote the strong chromatic index by $χ_{s}'(G)$ which is the minimum number of colors that allow a strong edge-coloring of $G$. Erdős and Nešetřil conjectured in 1985 that the upper bound of $χ_{s}'(G)$ is $\frac{5}{4}Δ^{2}$ when $Δ$ is even and $\frac{1}{4}(5Δ^{2}-2Δ+1)$ when $Δ$ is odd, where $Δ$ is the maximum degree of $G$. The conjecture is proved right when $Δ\leq3$. The best known upper bound for $Δ=4$ is 22 due to Cranston previously. In this paper we extend the result of Cranston to list strong edge-coloring, that is to say, we prove that when $Δ=4$ the upper bound of list strong chromatic index is 22.

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arXiv2017

Antimagic orientations of even regular graphs

A labeling of a digraph $D$ with $m$ arcs is a bijection from the set of arcs of $D$ to $\{1, \ldots, m\}$. A labeling of $D$ is antimagic if no two vertices in $D$ have the same vertex-sum, where the vertex-sum of a vertex $u\in V(D)$ for a labeling is the sum of labels of all arcs entering $u$ minus the sum of labels of all arcs leaving $u$. Motivated by the conjecture of Hartsfield and Ringel from 1990 on antimagic labelings of graphs, Hefetz, Mütze, and Schwartz [On antimagic directed graphs, J Graph Theory 64 (2010) 219--232] initiated the study of antimagic labelings of digraphs, and conjectured that every connected graph admits an antimagic orientation, where an orientation $D$ of a graph $G$ is antimagic if $D$ has an antimagic labeling. It remained unknown whether every disjoint union of cycles admits an antimagic orientation. In this paper, we first answer this question in the positive by proving that every $2$-regular graph has an antimagic orientation. We then show that for any integer $d\ge2$, every connected, $2d$-regular graph has an antimagic orientation. Our technique is new.

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