arXiv subjects

Binlong Li

Publications and source records attributed to Binlong Li.

At least 37 records · Page 2Linked to original sources

arXiv2021

The Turan problems of directed paths and cycles in digraphs

Let $\overrightarrow{P_k}$ and $\overrightarrow{C_k}$ denote the directed path and the directed cycle of order $k$, respectively. In this paper, we determine the precise maximum size of $\overrightarrow{P_k}$-free digraphs of order $n$ as well as the extremal digraphs attaining the maximum size for large $n$. For all $n$, we also determine the precise maximum size of $\overrightarrow{C_k}$-free digraphs of order $n$ as well as the extremal digraphs attaining the maximum size. In addition, Huang and Lyu [\textit{Discrete Math. 343(5) 2020}] characterized the extremal digraphs avoiding an orientation of $C_4$. For all other orientations of $C_4$, we also study the maximum size and the extremal digraphs avoiding them.

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arXiv2021

The stability method, eigenvalues and cycles of consecutive lengths

Woodall proved that for a graph $G$ of order $n\geq 2k+3$ where $k\geq 0$ is an integer, if $e(G)\geq \binom{n-k-1}{2}+\binom{k+2}{2}+1$ then $G$ contains a $C_{\ell}$ for each $\ell\in [3,n-k]$. In this article, we prove a stability result of this theorem. As a byproduct, we give complete solutions to two problems in \cite{GN19}. Our second part is devoted to an open problem by Nikiforov: what is the maximum $C$ such that for all positive $\varepsilon \sqrt{\lfloor\frac{n^2}{4}\rfloor}$ contains a cycle of length $\ell$ for every $\ell\leq (C-\varepsilon)n$. We prove that $C\geq\frac{1}{4}$ by a method different from previous ones, improving the existing bounds. We also derive an Erdős-Gallai type edge number condition for even cycles, which may be of independent interest.

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arXiv2020

A Strengthening of Erdős-Gallai Theorem and Proof of Woodall's Conjecture

For a 2-connected graph $G$ on $n$ vertices and two vertices $x,y\in V(G)$, we prove that there is an $(x,y)$-path of length at least $k$ if there are at least $\frac{n-1}{2}$ vertices in $V(G)\backslash \{x,y\}$ of degree at least $k$. This strengthens a well-known theorem due to Erdős and Gallai in 1959. As the first application of this result, we show that a 2-connected graph with $n$ vertices contains a cycle of length at least $2k$ if it has at least $\frac{n}{2}+k$ vertices of degree at least $k$. This confirms a 1975 conjecture made by Woodall. As another applications, we obtain some results which generalize previous theorems of Dirac, Erdős-Gallai, Bondy, and Fujisawa et al., present short proofs of the path case of Loebl-Komlós-Sós Conjecture which was verified by Bazgan et al. and of a conjecture of Bondy on longest cycles (for large graphs) which was confirmed by Fraisse and Fournier, and make progress on a conjecture of Bermond.

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arXiv2020

The anti-Ramsey number of $C_{3}$ and $C_{4}$ in the complete $r$-partite graphs

A subgraph of an edge-colored graph is rainbow, if all of its edges have different colors. For a graph $G$ and a family $\mathcal{H}$ of graphs, the anti-Ramsey number $ar(G, \mathcal{H})$ is the maximum number $k$ such that there exists an edge-coloring of $G$ with exactly $k$ colors without rainbow copy of any graph in $\mathcal{H}$. In this paper, we study the anti-Ramsey number of $C_{3}$ and $C_{4}$ in the complete $r$-partite graphs. For $r\ge 3$ and $n_{1}\ge n_{2}\ge \cdots\ge n_{r}\ge 1$, we determine $ ar(K_{n_{1}, n_{2}, \ldots, n_{r}},\{C_{3}, C_{4}\}), ar(K_{n_{1}, n_{2}, \ldots, n_{r}}, C_{3})$ and $ar(K_{n_{1}, n_{2}, \ldots, n_{r}}, C_{4})$.

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arXiv2020

Lumos: A Library for Diagnosing Metric Regressions in Web-Scale Applications

Web-scale applications can ship code on a daily to weekly cadence. These applications rely on online metrics to monitor the health of new releases. Regressions in metric values need to be detected and diagnosed as early as possible to reduce the disruption to users and product owners. Regressions in metrics can surface due to a variety of reasons: genuine product regressions, changes in user population, and bias due to telemetry loss (or processing) are among the common causes. Diagnosing the cause of these metric regressions is costly for engineering teams as they need to invest time in finding the root cause of the issue as soon as possible. We present Lumos, a Python library built using the principles of AB testing to systematically diagnose metric regressions to automate such analysis. Lumos has been deployed across the component teams in Microsoft's Real-Time Communication applications Skype and Microsoft Teams. It has enabled engineering teams to detect 100s of real changes in metrics and reject 1000s of false alarms detected by anomaly detectors. The application of Lumos has resulted in freeing up as much as 95% of the time allocated to metric-based investigations. In this work, we open source Lumos and present our results from applying it to two different components within the RTC group over millions of sessions. This general library can be coupled with any production system to manage the volume of alerting efficiently.

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arXiv2019

Faithful subgraphs and Hamiltonian circles of infinite graphs

A circle of an infinite locally finite graph $G$ is the imagine of a homeomorphic mapping of the unit circle $S^1$ in $|G|$, the Freudenthal compactification of $G$. A circle of $G$ is Hamiltonian if it meets every vertex (and then every end) of $G$. In this paper, we study a method for finding Hamiltonian circles of graphs. We illustrate this by extending several results on finite graphs to Hamiltonian circles in infinite graphs. For example, we prove that the prism of every 3-connected cubic graph has a Hamiltonian circle, extending the result of the finite case by Paulraja.

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arXiv2019

Hamiltonicity of bi-power of bipartite graphs, for finite and infinite cases

For a graph $G$, the $t$-th power $G^t$ is the graph on $V(G)$ such that two vertices are adjacent if and only if they have distance at most $t$ in $G$; and the $t$-th bi-power $G_B^t$ is the graph on $V(G)$ such that two vertices are adjacent if and only if their distance in $G$ is odd at most $t$. Fleischner's theorem states that the square of every 2-connected finite graph has a Hamiltonian cycle. Georgakopoulos prove that the square of every 2-connected infinite locally finite graph has a Hamiltonian circle. In this paper, we consider the Hamiltonicity of the bi-power of bipartite graphs. We show that for every connected finite bipartite graph $G$ with a perfect matching, $G_B^3$ has a Hamiltonian cycle. We also show that if $G$ is a connected infinite locally finite bipartite graph with a perfect matching, then $G_B^3$ has a Hamiltonian circle.

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arXiv2019

Exact bipartite Turán numbers of large even cycles

Let the bipartite Turán number $ex(m,n,H)$ of a graph $H$ be the maximum number of edges in an $H$-free bipartite graph with two parts of sizes $m$ and $n$, respectively. In this paper, we prove that $ex(m,n,C_{2t})=(t-1)n+m-t+1$ for any positive integers $m,n,t$ with $n\geq m\geq t\geq \frac{m}{2}+1$. This confirms the rest of a conjecture of Györi \cite{G97} (in a stronger form), and improves the upper bound of $ex(m,n,C_{2t})$ obtained by Jiang and Ma \cite{JM18} for this range. We also prove a tight edge condition for consecutive even cycles in bipartite graphs, which settles a conjecture in \cite{A09}. As a main tool, for a longest cycle $C$ in a bipartite graph, we obtain an estimate on the upper bound of the number of edges which are incident to at most one vertex in $C$. Our two results generalize or sharpen a classical theorem due to Jackson \cite{J85} in different ways.

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arXiv2018

Uniform sets in a family with restricted intersections

Let $\mathcal{F}$ be a family of subsets of $[n]=\{1,\ldots,n\}$ and let $L$ be a set of nonnegative integers. The family $\mathcal{F}$ is \emph{$L$-intersecting} if $|F\cap F'|\in L$ for every two distinct members $F,F'\in\mathcal{F}$; and $\mathcal{F}$ is $k$-uniform if all its members have the same size $k$. A large variety of problems and results in extremal set theory concern on $k$-uniform $L$-intersecting families. Many attentions are paid to finding the maximum size of a family among all $k$-uniform $L$-intersecting families with prescribed $n,k$ and $L$. In this paper, from another point of view, we propose and investigate the problem of estimating the maximum size of a member in a family among all uniform $L$-intersecting families with size $m$, here $n,m$ and $L$ are prescribed. Our results aim to find out more precise relations of $n,m,k$ and $L$.

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arXiv2018

Connected $k$-factors in bipartite graphs

Let $k,l$ be two positive integers. An $S_{k,l}$ is a graph obtained from disjoint $K_{1,k}$ and $K_{1,l}$ by adding an edge between the $k$-degree vertex in $K_{1,k}$ and the $l$-degree vertex in $K_{1,l}$. An {\em $S_{k,l}$-free} graph is a graph containing no induced subgraph isomorphic to $S_{k,l}$. In this note, we show that, for any positive integers $k,l$ with $2\leqslant k\leqslant l$, there exists a constant $c=c(k,l)$ such that every connected balanced $S_{k,l}$-free bipartite graph with minimum degree at least $c$ contains a connected $k$-factor.

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arXiv2018

Extremal problems on the Hamiltonicity of claw-free graphs

In 1962, Erdős proved that if a graph $G$ with $n$ vertices satisfies $$ e(G)>\max\left\{\binom{n-k}{2}+k^2,\binom{\lceil(n+1)/2\rceil}{2}+\left\lfloor \frac{n-1}{2}\right\rfloor^2\right\}, $$ where the minimum degree $δ(G)\geq k$ and $1\leq k\leq(n-1)/2$, then it is Hamiltonian. For $n \geq 2k+1$, let $E^k_n=K_{k}\vee (kK_1+K_{n-2k})$, where "$\vee$" is the "join" operation. One can observe $e(E^k_n)=\binom{n-k}{2}+k^2$ and $E^k_n$ is not Hamiltonian. As $E^k_n$ contains induced claws for $k\geq 2$, a natural question is to characterize all 2-connected claw-free non-Hamiltonian graphs with the largest possible number of edges. We answer this question completely by proving a claw-free analog of Erdős' theorem. Moreover, as byproducts, we establish several tight spectral conditions for a 2-connected claw-free graph to be Hamiltonian. Similar results for the traceability of connected claw-free graphs are also obtained. Our tools include Ryjáček's claw-free closure theory and Brousek's characterization of minimal 2-connected claw-free non-Hamiltonian graphs.

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arXiv2018

Kernels by rainbow paths in arc-colored tournaments

For an arc-colored digraph $D$, define its {\em kernel by rainbow paths} to be a set $S$ of vertices such that (i) no two vertices of $S$ are connected by a rainbow path in $D$, and (ii) every vertex outside $S$ can reach $S$ by a rainbow path in $D$. In this paper, we show that it is NP-complete to decide whether an arc-colored tournament has a kernel by rainbow paths, where a {\em tournament} is an orientation of a complete graph. In addition, we show that every arc-colored $n$-vertex tournament with all its strongly connected $k$-vertex subtournaments, $3\leq k\leq n$, colored with at least $k-1$ colors has a kernel by rainbow paths, and the number of colors required cannot be reduced.

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arXiv2016

Spectral analogues of Moon-Moser's theorem on Hamilton paths in bipartite graphs

In 1962, Erdős proved a theorem on the existence of Hamilton cycles in graphs with given minimum degree and number of edges. Significantly strengthening in case of balanced bipartite graphs, Moon and Moser proved a corresponding theorem in 1963. In this paper we establish several spectral analogues of Moon and Moser's theorem on Hamilton paths in balanced bipartite graphs and nearly balanced bipartite graphs. One main ingredient of our proofs is a structural result of its own interest, involving Hamilton paths in balanced bipartite graphs with given minimum degree and number of edges.

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arXiv2016

Heavy subgraphs, stability and hamiltonicity

Let $G$ be a graph. Adopting the terminology of Broersma et al. and Čada, respectively, we say that $G$ is 2-heavy if every induced claw ($K_{1,3}$) of $G$ contains two end-vertices each one has degree at least $|V(G)|/2$; and $G$ is o-heavy if every induced claw of $G$ contains two end-vertices with degree sum at least $|V(G)|$ in $G$. In this paper, we introduce a new concept, and say that $G$ is \emph{$S$-c-heavy} if for a given graph $S$ and every induced subgraph $G'$ of $G$ isomorphic to $S$ and every maximal clique $C$ of $G'$, every non-trivial component of $G'-C$ contains a vertex of degree at least $|V(G)|/2$ in $G$. In terms of this concept, our original motivation that a theorem of Hu in 1999 can be stated as every 2-connected 2-heavy and $N$-c-heavy graph is hamiltonian, where $N$ is the graph obtained from a triangle by adding three disjoint pendant edges. In this paper, we will characterize all connected graphs $S$ such that every 2-connected o-heavy and $S$-c-heavy graph is hamiltonian. Our work results in a different proof of a stronger version of Hu's theorem. Furthermore, our main result improves or extends several previous results.

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arXiv2016

Spectral analogues of Erdős' and Moon-Moser's theorems on Hamilton cycles

In 1962, Erdős gave a sufficient condition for Hamilton cycles in terms of the vertex number, edge number, and minimum degree of graphs which generalized Ore's theorem. One year later, Moon and Moser gave an analogous result for Hamilton cycles in balanced bipartite graphs. In this paper we present the spectral analogues of Erdős' theorem and Moon-Moser's theorem, respectively. Let $\mathcal{G}_n^k$ be the class of non-Hamiltonian graphs of order $n$ and minimum degree at least $k$. We determine the maximum (signless Laplacian) spectral radius of graphs in $\mathcal{G}_n^k$ (for large enough $n$), and the minimum (signless Laplacian) spectral radius of the complements of graphs in $\mathcal{G}_n^k$. All extremal graphs with the maximum (signless Laplacian) spectral radius and with the minimum (signless Laplacian) spectral radius of the complements are determined, respectively. We also solve similar problems for balanced bipartite graphs and the quasi-complements.

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arXiv2016

Induced subgraphs with large degrees at end-vertices for hamiltonicity of claw-free graphs

A graph is called \emph{claw-free} if it contains no induced subgraph isomorphic to $K_{1,3}$. Matthews and Sumner proved that a 2-connected claw-free graph $G$ is hamiltonian if every vertex of it has degree at least $(|V(G)|-2)/3$. At the workshop C\&C (Novy Smokovec, 1993), Broersma conjectured the degree condition of this result can be restricted only to end-vertices of induced copies of $N$ (the graph obtained from a triangle by adding three disjoint pendant edges). Fujisawa and Yamashita showed that the degree condition of Matthews and Sumner can be restricted only to end-vertices of induced copies of $Z_1$ (the graph obtained from a triangle by adding one pendant edge). Our main result in this paper is a characterization of all graphs $H$ such that a 2-connected claw-free graph $G$ is hamiltonian if each end-vertex of every induced copy of $H$ in $G$ has degree at least $|V(G)|/3+1$. This gives an affirmative solution of the conjecture of Broersma up to an additive constant.

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arXiv2016

Disjoint cycles of different lengths in graphs and digraphs

Understanding how the cycles of a graph or digraph behave in general has always been an important point of graph theory. In this paper, we study the question of finding a set of $k$ vertex-disjoint cycles (resp. directed cycles) of distinct lengths in a given graph (resp. digraph). In the context of undirected graphs, we prove that, for every $k \geq 1$, every graph with minimum degree at least $\frac{k^2+5k-2}{2}$ has $k$ vertex-disjoint cycles of different lengths, where the degree bound is best possible. We also consider stronger situations, and exhibit degree bounds (some of which are best possible) when e.g. the graph is triangle-free, or the $k$ cycles are requested to have different lengths congruent to some values modulo some $r$. In the context of directed graphs, we consider a conjecture of Lichiardopol concerning the least minimum out-degree required for a digraph to have $k$ vertex-disjoint directed cycles of different lengths. We verify this conjecture for tournaments, and, by using the probabilistic method, for regular digraphs and digraphs of small order.

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arXiv2015

Solution to a problem on hamiltonicity of graphs under Ore- and Fan-type heavy subgraph conditions

A graph $G$ is called \emph{claw-o-heavy} if every induced claw ($K_{1,3}$) of $G$ has two end-vertices with degree sum at least $|V(G)|$ in $G$. For a given graph $R$, $G$ is called \emph{$R$-f-heavy} if for every induced subgraph $H$ of $G$ isomorphic to $R$ and every pair of vertices $u,v\in V(H)$ with $d_H(u,v)=2$, there holds $\max\{d(u),d(v)\}\geq |V(G)|/2$. In this paper, we prove that every 2-connected claw-\emph{o}-heavy and $Z_3$-\emph{f}-heavy graph is hamiltonian (with two exceptional graphs), where $Z_3$ is the graph obtained from identifying one end-vertex of $P_4$ (a path with 4 vertices) with one vertex of a triangle. This result gives a positive answer to a problem proposed in [B. Ning, S. Zhang, Ore- and Fan-type heavy subgraphs for Hamiltonicity of 2-connected graphs, Discrete Math. 313 (2013) 1715--1725], and also implies two previous theorems of Faudree et al. and Chen et al., respectively.

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