arXiv subjects

Xingxing Yu

Publications and source records attributed to Xingxing Yu.

At least 37 records · Page 2Linked to original sources

arXiv2021

A note on exact minimum degree threshold for fractional perfect matchings

Rödl, Ruciński, and Szemerédi determined the minimum $(k-1)$-degree threshold for the existence of fractional perfect matchings in $k$-uniform hypergrahs, and Kühn, Osthus, and Townsend extended this result by asymptotically determining the $d$-degree threshold for the range $k-1>d\ge k/2$. In this note, we prove the following exact degree threshold: Let $k,d$ be positive integers with $k\ge 4$ and $k-1>d\geq k/2$, and let $n$ be any integer with $n\ge k^2$. Then any $n$-vertex $k$-uniform hypergraph with minimum $d$-degree $δ_d(H)>{n-d\choose k-d} -{n-d-(\lceil n/k\rceil-1)\choose k-d}$ contains a fractional perfect matching. This lower bound on the minimum $d$-degree is best possible. We also determine optimal minimum $d$-degree conditions which guarantees the existence of fractional matchings of size $s$, where $0<s\le n/k$ (when $k/2\le d\le k-1$), or with $s$ large enough and $s\le n/k$ (when $2k/5<d<k/2$).

math.CO↗

arXiv2021

A better bound on the size of rainbow matchings

Aharoni and Howard conjectured that, for positive integers $n,k,t$ with $n\ge k$ and $n\ge t$, if $F_1,\ldots, F_t\subseteq {[n]\choose k}$ such that $|F_i|>{n\choose k}-{n-t+1\choose k}$ for $i\in [t]$ then there exist $e_i\in F_i$ for $i\in [t]$ such that $e_1,\ldots,e_t$ are pairwise disjoint. Huang, Loh, and Sudakov proved this conjecture for $t<n/(3k^2)$. In this paper, we show that this conjecture holds for $t\le n/(2k)$ and $n$ sufficiently large.

math.CO↗

arXiv2020

Partitioning digraphs with outdegree at least 4

Scott asked the question of determining $c_d$ such that if $D$ is a digraph with $m$ arcs and minimum outdegree $d\ge 2$ then $V(D)$ has a partition $V_1, V_2$ such that $\min\left\{e(V_1,V_2),e(V_2, V_1)\right\}\geq c_dm$, where $e(V_1,V_2)$ (respectively, $e(V_2,V_1)$) is the number of arcs from $V_1$ to $V_2$ (respectively, from $V_2$ to $V_1$). Lee, Loh, and Sudakov showed that $c_2=1/6+o(1)$ and $c_3=1/5+o(1)$, and conjectured that $c_d= \frac{d-1}{2(2d-1)}+o(1)$ for $d\ge 4$. In this paper, we show $c_4=3/14+o(1)$ and prove some partial results for $d\ge 5$.

math.CO↗

arXiv2020

Rainbow matchings for 3-uniform hypergraphs

Kühn, Osthus, and Treglown and, independently, Khan proved that if $H$ is a $3$-uniform hypergraph with $n$ vertices such that $n\in 3\mathbb{Z}$ and large, and $δ_1(H)>{n-1\choose 2}-{2n/3\choose 2}$, then $H$ contains a perfect matching. In this paper, we show that for $n\in 3\mathbb{Z}$ sufficiently large, if $F_1, \ldots, F_{n/3}$ are 3-uniform hypergrapghs with a common vertex set and $δ_1(F_i)>{n-1\choose 2}-{2n/3\choose 2}$ for $i\in [n/3]$, then $\{F_1,\dots, F_{n/3}\}$ admits a rainbow matching, i.e., a matching consisting of one edge from each $F_i$. This is done by converting the rainbow matching problem to a perfect matching problem in a special class of uniform hypergraphs.

math.CO↗

arXiv2020

4-Separations in Hajós Graphs

As a natural extension of the Four Color Theorem, Hajós conjectured that graphs containing no $K_5$-subdivision are 4-colorable. Any possible counterexample to this conjecture with minimum number of vertices is called a {\it Hajós graph}. Previous results show that Hajós graphs are 4-connected but not 5-connected. A $k$-separation in a graph $G$ is a pair $(G_1,G_2)$ of edge-disjoint subgraphs of $G$ such that $|V(G_1\cap G_2)|=k$, $G=G_1\cup G_2$, and $G_i\not\subseteq G_{3-i}$ for $i=1,2$. In this paper, we show that Hajós graphs do not admit a 4-separation $(G_1,G_2)$ such that $|V(G_1)|\ge 6$ and $G_1$ can be drawn in the plane with no edge crossings and all vertices in $V(G_1\cap G_2)$ incident with a common face. This is a step in our attempt to reduce Hajós' conjecture to the Four Color Theorem.

math.CO↗

arXiv2020

Large cycles in essentially 4-connected graphs

Tutte proved that every 4-connected planar graph contains a Hamilton cycle, but there are 3-connected $n$-vertex planar graphs whose longest cycles have length $Θ(n^{\log_32})$. On the other hand, Jackson and Wormald in 1992 proved that an essentially 4-connected $n$-vertex planar graph contains a cycle of length at least $(2n+4)/5$, which was recently improved to $5(n+2)/8$ by Fabrici {\it et al}. In this paper, we improve this bound to $\lceil (2n+6)/3\rceil$ for $n\ge 6$, which is best possible, by proving a quantitative version of a result of Thomassen on Tutte paths.

math.CO↗

arXiv2019

Monochromatic subgraphs in iterated triangulations

For integers $n\ge 0$, an iterated triangulation $Tr(n)$ is defined recursively as follows: $Tr(0)$ is the plane triangulation on three vertices and, for $n\ge 1$, $Tr(n)$ is the plane triangulation obtained from the plane triangulation $Tr(n-1)$ by, for each inner face $F$ of $Tr(n-1)$, adding inside $F$ a new vertex and three edges joining this new vertex to the three vertices incident with $F$. In this paper, we show that there exists a 2-edge-coloring of $Tr(n)$ such that $Tr(n)$ contains no monochromatic copy of the cycle $C_k$ for any $k\ge 5$. As a consequence, the answer to one of two questions asked by Axenovich, Schade, Thomassen and Ueckerdt is negative. We also determine the radius two graphs $H$ for which there exists $n$ such that every 2-edge-coloring of $Tr(n)$ contains a monochromatic copy of $H$, extending a result of the above authors for radius two trees.

math.CO↗

arXiv2019

Wheels in planar graphs and Hajós graphs

It was conjectured by Hajós that graphs containing no $K_5$-subdivision are 4-colorable. Previous results show that any possible minimum counterexample to Hajós' conjecture, called Hajós graph, is 4-connected but not 5-connected. In this paper, we show that if a Hajós graph admits a 4-cut or 5-cut with a planar side then the planar side must be small or contains a special wheel. This is a step in our effort to reduce Hajós' conjecture to the Four Color Theorem.

math.CO↗

arXiv2019

Near perfect matchings in uniform hypergraphs

In this paper, we study degree conditions for the existence of large matchings in uniform hypergraphs. We prove that for integers $k,l,n$ with $k\ge 3$, $k/2 {n-l\choose k-l}-{(n-l)-(\lceil n/k \rceil-2)\choose 2}$, then $H$ has a matching covering all but a constant number of vertices. When $l=k-2$ and $k\ge 5$, such a matching is near perfect and our bound on $δ_l(H)$ is best possible. When $k=3$, with the help of an absorbing lemma of Hán, Person, and Schacht, our proof also implies that $H$ has a perfect matching, a result proved by K\" uhn, Osthus, and Treglown and, independently, of Kahn.

math.CO↗

arXiv2019

7-Connected Graphs are 4-Ordered

A graph $G$ is $k$-ordered if for any distinct vertices $v_1, v_2, \ldots, v_k \in V(G)$, it has a cycle through $v_1, v_2, \ldots, v_k$ in order. Let $f(k)$ denote the minimum integer so that every $f(k)$-connected graph is $k$-ordered. The first non-trivial case of determining $f(k)$ is when $k=4$, where the previously best known bounds are $7 \leq f(4) \leq 40$. We prove that in fact $f(4)=7$.

math.CO↗

arXiv2018

A bound on judicious bipartitions of directed graphs

Judicious partitioning problems on graphs ask for partitions that bound several quantities simultaneously, which have received a lot of attentions lately. Scott asked the following natural question: What is the maximum constant $c_d$ such that every directed graph $D$ with $m$ arcs and minimum outdegree $d$ admits a bipartition $V(D)= V_1\cup V_2$ satisfying $\min\{e(V_1, V_2), e(V_2, V_1)\}\ge c_d m$? Here, for $i=1,2$, $e(V_{i},V_{3-i})$ denotes the number of arcs in $D$ from $V_{i}$ to $V_{3-i}$. Lee, Loh, and Sudakov %[Judicious partitions of directed graphs, Random Struct. Alg. 48 %(2016) 147--170] conjectured that every directed graph $D$ with $m$ arcs and minimum outdegree at least $d\ge 2$ admits a bipartition $V(D)=V_1\cup V_2$ such that \[ \min\{e(V_1,V_2),e(V_2,V_1)\}\geq \Big(\frac{d-1}{2(2d-1)}+ o(1)\Big)m. \] %While it is not known whether or not the minimum outdegree condition %alone is sufficient, w We show that this conjecture holds under the additional natural condition that the minimum indegree is also at least $d$.

math.CO↗

arXiv2018

Graph-Based Radio Resource Management for Vehicular Networks

This paper investigates the resource allocation problem in device-to-device (D2D)-based vehicular communications, based on slow fading statistics of channel state information (CSI), to alleviate signaling overhead for reporting rapidly varying accurate CSI of mobile links. We consider the case when each vehicle-to-infrastructure (V2I) link shares spectrum with multiple vehicle-to-vehicle (V2V) links. Leveraging the slow fading statistical CSI of mobile links, we maximize the sum V2I capacity while guaranteeing the reliability of all V2V links. We propose a graph-based algorithm that uses graph partitioning tools to divide highly interfering V2V links into different clusters before formulating the spectrum sharing problem as a weighted 3-dimensional matching problem, which is then solved through adapting a high-performance approximation algorithm.

cs.IT↗

arXiv2017

Minimum co-degree condition for perfect matchings in k-partite k-graphs

Let $H$ be a $k$-partite $k$-graph with $n$ vertices in each partition class, and let $δ_{k-1}(H)$ denote the minimum co-degree of $H$. We characterize those $H$ with $δ_{k-1}(H) \geq n/2$ and with no perfect matching. As a consequence we give an affirmative answer to the following question of Rödl and Ruciński: If $k$ is even or $n \not\equiv 2 \pmod 4$, does $δ_{k-1}(H) \geq n/2$ imply that $H$ has a perfect matching? We also give an example indicating that it is not sufficient to impose this degree bound on only two types of $(k-1)$-sets.

math.CO↗

arXiv2017

Circumference of 3-connected cubic graphs

The circumference of a graph is the length of its longest cycles. Jackson established a conjecture of Bondy by showing that the circumference of a 3-connected cubic graph of order $n$ is $Ω(n^{0.694})$. Bilinski {\it et al.} improved this lower bound to $Ω(n^{0.753})$ by studying large Eulerian subgraphs in 3-edge-connected graphs. In this paper, we further improve this lower bound to $Ω(n^{0.8})$. This is done by considering certain 2-connected cubic graphs, finding cycles through two given edges, and distinguishing the cases whether or not these edges are adjacent.

math.CO↗

arXiv2017

Almost perfect matchings in $k$-partite $k$-graphs

The minimum co-degree threshold for a perfect matching in a $k$-graph with $n$ vertices was determined by Rödl, Ruciński and Szemerédi for the case when $n\equiv 0\pmod k$. Recently, Han resolved the remaining cases when $n \not\equiv 0\pmod k$, establishing a conjecture of Rödl, Ruciński and Szemerédi. In this paper, we determine the minimum co-degree threshold for almost perfect matchings in $k$-partite $k$-graphs, answering a question of Rödl and Ruciński.

math.CO↗

arXiv2017

On problems about judicious bipartitions of graphs

Bollobás and Scott [5] conjectured that every graph $G$ has a balanced bipartite spanning subgraph $H$ such that for each $v\in V(G)$, $d_H(v)\ge (d_G(v)-1)/2$. In this paper, we show that every graphic sequence has a realization for which this Bollobás-Scott conjecture holds, confirming a conjecture of Hartke and Seacrest [10]. On the other hand, we give an infinite family of counterexamples to this Bollobás-Scott conjecture, which indicates that $\lfloor (d_G(v)-1)/2\rfloor$ (rather than $(d_G(v)-1)/2$) is probably the correct lower bound. We also study bipartitions $V_1, V_2$ of graphs with a fixed number of edges. We provide a (best possible) upper bound on $e(V_1)^λ+e(V_2)^λ$ for any real $λ\geq 1$ (the case $λ=2$ is a question of Scott [13]) and answer a question of Scott [13] on $\max\{e(V_1),e(V_2)\}$.

math.CO↗

arXiv2016

The Kelmans-Seymour conjecture IV: a proof

A well known theorem of Kuratowski in 1932 states that a graph is planar if, and only if, it does not contain a subdivision of $K_5$ or $K_{3,3}$. Wagner proved in 1937 that if a graph other than $K_5$ does not contain any subdivision of $K_{3,3}$ then it is planar or it admits a cut of size at most 2. Kelmans and, independently, Seymour conjectured in the 1970s that if a graph does not contain any subdivision of $K_5$ then it is planar or it admits a cut of size at most 4. In this paper, we give a proof of the Kelmans-Seymour conjecture. We also discuss several related results and problems.

math.CO↗

arXiv2016

On rainbow matchings for hypergraphs

For any posotive integer $m$, let $[m]:=\{1,\ldots,m\}$. Let $n,k,t$ be positive integers. Aharoni and Howard conjectured that if, for $i\in [t]$, $\mathcal{F}_i\subset[n]^k:= \{(a_1,\ldots,a_k): a_j\in [n] \mbox{ for } j\in [k]\}$ and $|\mathcal{F}_i|>(t-1)n^{k-1}$, then there exist $M\subseteq [n]^k$ such that $|M|=t$ and $|M\cap \mathcal{F}_i|=1$ for $i\in [t]$ We show that this conjecture holds when $n\geq 3(k-1)(t-1)$. Let $n, t, k_1\ge k_2\geq \ldots\geq k_t $ be positive integers. Huang, Loh and Sudakov asked for the maximum $Π_{i=1}^t |{\cal R}_i|$ over all ${\cal R}=\{{\cal R}_1, \ldots ,{\cal R}_t\}$ such that each ${\cal R}_i$ is a collection of $k_i$-subsets of $[n]$ for which there does not exist a collection $M$ of subsets of $[n]$ such that $|M|=t$ and $|M\cap \mathcal{R}_i|=1$ for $i\in [t]$ %and ${\cal R}$ does not admit a rainbow matching. We show that for sufficiently large $n$ with $\sum_{i=1}^t k_i\leq n(1-(4k\ln n/n)^{1/k}) $, $\prod_{i=1}^t |\mathcal{R}_i|\leq {n-1\choose k_1-1}{n-1\choose k_2-1}\prod_{i=3}^{t}{n\choose k_i}$. This bound is tight.

math.CO↗