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Let $m,n,r,s$ be nonnegative integers such that $n\ge m=3r+s$ and $1\leq s\leq 3$. Let \[\delta(n,r,s)=\left\{\begin{array}{ll} n^2-(n-r)^2 &\text{if}\ s=1 , \\[5pt] n^2-(n-r+1)(n-r-1) &\text{if}\ s=2,\\[5pt] n^2 - (n-r)(n-r-1) &\text{if}\ s=3. \end{array}\right.\] We show that there exists a constant $n_0 > 0$ such that if $F_1,\ldots, F_n$ are 3-partite 3-graphs with $n\ge n_0$ vertices in each partition class and minimum vertex degree of $F_i$ is at least $\delta(n,r,s)+1$ for $i \in [n]$ then $\{F_1,\ldots,F_n\}$ admits a rainbow perfect matching. This generalizes a result of Lo and Markstr\"om on the vertex degree threshold for the existence of perfect matchings in 3-partite 3-graphs. In this proof, we use a fractional rainbow matching theory obtained by Aharoni et al. to find edge-disjoint fractional perfect matching.

In this paper, we determine the minimum degree threshold of perfect matchings with high discrepancy in $r$-edge-colored $k$-uniform hypergraphs for all $k\geq 3$ and $r\geq 2$, thereby completing the investigation into discrepancies of perfect matchings that has recently attracted significant attention. Our approach identifies this discrepancy threshold with a novel family of multicolored uniform hypergraphs and reveals new phenomena not covered in previous studies. In particular, our results address a question of Balogh, Treglown and Z\'arate-Guer\'en concerning 3-uniform hypergraphs., Comment: 25 pages with 1 figure; added acknowledgements and a new appendix to address a missing case in the original Lemma 12

To alleviate energy shortages and environmental impacts caused by transportation, this study introduces EcoFollower, a novel eco-car-following model developed using reinforcement learning (RL) to optimize fuel consumption in car-following scenarios. Employing the NGSIM datasets, the performance of EcoFollower was assessed in comparison with the well-established Intelligent Driver Model (IDM). The findings demonstrate that EcoFollower excels in simulating realistic driving behaviors, maintaining smooth vehicle operations, and closely matching the ground truth metrics of time-to-collision (TTC), headway, and comfort. Notably, the model achieved a significant reduction in fuel consumption, lowering it by 10.42\% compared to actual driving scenarios. These results underscore the capability of RL-based models like EcoFollower to enhance autonomous vehicle algorithms, promoting safer and more energy-efficient driving strategies.

Recent years have seen a growing research interest in applications of Deep Neural Networks (DNN) on autonomous vehicle technology. The trend started with perception and prediction a few years ago and it is gradually being applied to motion planning tasks. Despite the performance of networks improve over time, DNN planners inherit the natural drawbacks of Deep Learning. Learning-based planners have limitations in achieving perfect accuracy on the training dataset and network performance can be affected by out-of-distribution problem. In this paper, we propose FusionAssurance, a novel trajectory-based end-to-end driving fusion framework which combines physics-informed control for safety assurance. By incorporating Potential Field into Model Predictive Control, FusionAssurance is capable of navigating through scenarios that are not included in the training dataset and scenarios where neural network fail to generalize. The effectiveness of the approach is demonstrated by extensive experiments under various scenarios on the CARLA benchmark.

Let $n,k,s$ be three integers and $\beta$ be a sufficiently small positive number such that $k\geq 3$, $0<1/n\ll \beta\ll 1/k$ and $ks+k\leq n\leq (1+\beta)ks+k-2$. A $k$-graph is called non-trivial if it has no isolated vertex. In this paper, we determine the maximum number of edges in a non-trivial $k$-graph with $n$ vertices and matching number at most $s$. This result confirms a conjecture proposed by Frankl (On non-trivial families without a perfect matching, \emph{European J. Combin.}, \textbf{84} (2020), 103044) for the case when $s$ is sufficiently large.

Let $G$ be a graph and $g,f:V(G)\to2^N$ be two set functions such that $g(v)\le f(v)$ and $g(v)\equiv f(v)\pmod 2$ for every $v\in V(G)$. An orientation $O$ of $G$ is called a $(g,f)$-parity orientation if $g(v)\le d^+_O(v)\le f(v)$ and $g(v)\equiv d^+_O(v)\pmod 2$ for every $v\in V(G)$. In this paper, we give a Tutte-type characterization for a graph to have a $(g,f)$-parity orientation.

A factor of a graph is a spanning subgraph satisfying some given conditions. An earlier survey of factors can be traced back to the Akiyama and Kano [J. Graph Theory, 1985, 9: 1-42] in which they described the characterization of factors in (bipartite) graphs and digraphs, respectively. Soon after, Kouider and Vestergaard summarized the findings related to connected factors [Graphs Combin., 2005, 21(1): 1-26]. Plummer extended the aforementioned research by providing a comprehensive overview of progress made in the study of graph factors and factorization from 1985 to 2003 [Discrete Math., 2007, 7-8: 791-821]. In this paper, we aim to summarize the relevant results regarding factors from the perspective of eigenvalues., Comment: Comments are very welcome!

Let $G$ be a graph and $F:V(G)\to2^N$ be a set function. The graph $G$ is said to be \emph{F-avoiding} if there exists an orientation $O$ of $G$ such that $d^+_O(v)\notin F(v)$ for every $v\in V(G)$, where $d^+_O(v)$ denotes the out-degree of $v$ in the directed graph $G$ with respect to $O$. In this paper, we give a Tutte-type good characterization to decide the $F$-avoiding problem when for every $v\in V(G)$, $|F(v)|\leq \frac{1}{2}(d_G(v)+1)$ and $F(v)$ contains no two consecutive integers. Our proof also gives a simple polynomial algorithm to find a desired orientation. As a corollary, we prove the following result: if for every $v\in V(G)$, $|F(v)|\leq \frac{1}{2}(d_G(v)+1)$ and $F(v)$ contains no two consecutive integers, then $G$ is $F$-avoiding. This partly answers a problem proposed by Akbari et. al.(2020)

Prediction, decision-making, and motion planning are essential for autonomous driving. In most contemporary works, they are considered as individual modules or combined into a multi-task learning paradigm with a shared backbone but separate task heads. However, we argue that they should be integrated into a comprehensive framework. Although several recent approaches follow this scheme, they suffer from complicated input representations and redundant framework designs. More importantly, they can not make long-term predictions about future driving scenarios. To address these issues, we rethink the necessity of each module in an autonomous driving task and incorporate only the required modules into a minimalist autonomous driving framework. We propose BEVGPT, a generative pre-trained large model that integrates driving scenario prediction, decision-making, and motion planning. The model takes the bird's-eye-view (BEV) images as the only input source and makes driving decisions based on surrounding traffic scenarios. To ensure driving trajectory feasibility and smoothness, we develop an optimization-based motion planning method. We instantiate BEVGPT on Lyft Level 5 Dataset and use Woven Planet L5Kit for realistic driving simulation. The effectiveness and robustness of the proposed framework are verified by the fact that it outperforms previous methods in 100% decision-making metrics and 66% motion planning metrics. Furthermore, the ability of our framework to accurately generate BEV images over the long term is demonstrated through the task of driving scenario prediction. To the best of our knowledge, this is the first generative pre-trained large model for autonomous driving prediction, decision-making, and motion planning with only BEV images as input., Comment: AAAI2024

Car-following is a control process in which a following vehicle (FV) adjusts its acceleration to keep a safe distance from the lead vehicle (LV). Recently, there has been a booming of data-driven models that enable more accurate modeling of car-following through real-world driving datasets. Although there are several public datasets available, their formats are not always consistent, making it challenging to determine the state-of-the-art models and how well a new model performs compared to existing ones. In contrast, research fields such as image recognition and object detection have benchmark datasets like ImageNet, Microsoft COCO, and KITTI. To address this gap and promote the development of microscopic traffic flow modeling, we establish a public benchmark dataset for car-following behavior modeling. The benchmark consists of more than 80K car-following events extracted from five public driving datasets using the same criteria. These events cover diverse situations including different road types, various weather conditions, and mixed traffic flows with autonomous vehicles. Moreover, to give an overview of current progress in car-following modeling, we implemented and tested representative baseline models with the benchmark. Results show that the deep deterministic policy gradient (DDPG) based model performs competitively with a lower MSE for spacing compared to traditional intelligent driver model (IDM) and Gazis-Herman-Rothery (GHR) models, and a smaller collision rate compared to fully connected neural network (NN) and long short-term memory (LSTM) models in most datasets. The established benchmark will provide researchers with consistent data formats and metrics for cross-comparing different car-following models, promoting the development of more accurate models. We open-source our dataset and implementation code in https://github.com/HKUST-DRIVE-AI-LAB/FollowNet.

Let $n,s,$ and $k$ be positive integers such that $k\geq 3$, $s\geq 3$ and $n\geq ks$. An $s$-matching $M_s$ in a $k$-uniform hypergraph is a set of $s$ pairwise disjoint edges. The anti-Ramsey number $\textrm{ar}(n,k,M_s)$ of an $s$-matching is the smallest integer $c$ such that each edge-coloring of the $n$-vertex $k$-uniform complete hypergraph with exactly $c$ colors contains an $s$-matching with distinct colors. In 2013, \"Ozkahya and Young proposed a conjecture on the exact value of ar$(n,k,M_s)$ for all $n \geq sk$ and $k \geq 3$. A 2019 result by Frankl and Kupavskii verified this conjecture for all $n \geq sk+(s-1)(k-1)$ and $k \geq 3$. We aim to determine the value of ar$(n,3,M_s)$ for $3s \leq n < 5s-2$ in this paper. Namely, we prove that if $3s

We show that for any integer $k\ge 1$ there exists an integer $t_0(k)$ such that for integers $t, k_1, \ldots, k_{t+1}, n$ with $t>t_0(k)$, $\max\{k_1, \ldots, k_{t+1}\}\le k$, and $n > 2k(t+1)$, the following holds: If $F_i \subseteq {[n]\choose k_i}$ and $|F_i|> {n\choose k_i}-{n-t\choose k_i} - {n-t-k \choose k_i-1} + 1$ for all $i \in [t+1]$, then either $\{F_1,\ldots, F_{t+1}\}$ admits a rainbow matching of size $t+1$ or there exists $W\in {[n]\choose t}$ such that $W$ is a vertex cover of $F_i$ for all $i\in [t+1]$. This may be viewed as a rainbow non-uniform extension of the classical Hilton-Milner theorem. We also show that the same holds for every $t$ and $n > 2k^3t$, generalizing a recent stability result of Frankl and Kupavskii on matchings to rainbow matchings., Comment: arXiv admin note: text overlap with arXiv:2004.12561

A 3-graph $\mathcal{F}$ is \emph{$U(s, 2s+1)$} if for any $s$ edges $e_1,...,e_s\in E(\mathcal{F})$, $|e_1\cup...\cup e_s|\leq 2s+1$. Frankl and Kupavskii (2020) proposed the following conjecture: For any $3$-graph $\mathcal{F}$ with $n$ vertices, if $\mathcal{F}$ is $U(s, 2s+1)$, then $$e(\mathcal{F})\leq \max\left\{{n-1\choose 2}, (n-s-1){s+1\choose 2}+{s+1\choose 3}, {2s+1\choose 3}\right\}.$$ In this paper, we confirm Frankl and Kupavskii's conjecture.

Let $n,m$ be integers such that $1\leq m\leq (n-2)/2$ and let $[n]=\{1,\ldots,n\}$. Let $\mathcal{G}=\{G_1,\ldots,G_{m+1}\}$ be a family of graphs on the same vertex set $[n]$. In this paper, we prove that if for any $i\in [m+1]$, the spectral radius of $G_i$ is not less than $\max\{2m,\frac{1}{2}(m-1+\sqrt{(m-1)^2+4m(n-m)})\}$, then $\mathcal{G}$ admits a rainbow matching, i.e. a choice of disjoint edges $e_i\in G_i$, unless $G_1=G_2=\ldots=G_{m+1}$ and $G_1\in \{K_{2m+1}\cup (n-2m-1)K_1, K_m\vee (n-m)K_1\}$.

The study of the existence of hamiltonian cycles in a graph is a classic problem in graph theory. By incorporating toughness and spectral conditions, we can consider Chv\'{a}tal's conjecture from another perspective: what is the spectral condition to guarantee the existence of a hamiltonian cycle among $t$-tough graphs? We first give the answer to $1$-tough graphs, i.e. if $\rho(G)\geq\rho(M_{n})$, then $G$ contains a hamiltonian cycle, unless $G\cong M_{n}$, where $M_{n}=K_{1}\nabla K_{n-4}^{+3}$ and $K_{n-4}^{+3}$ is the graph obtained from $3K_{1}\cup K_{n-4}$ by adding three independent edges between $3K_{1}$ and $K_{n-4}$. The Brouwer's toughness theorem states that every $d$-regular connected graph always has $t(G)>\frac{d}{\lambda}-1$ where $\lambda$ is the second largest absolute eigenvalue of the adjacency matrix. In this paper, we extend the result in terms of its spectral radius, i.e. we provide a spectral condition for a graph to be 1-tough with minimum degree $\delta$ and to be $t$-tough, respectively., Comment: 13 pages

An $[a,b]$-factor of a graph $G$ is a spanning subgraph $H$ such that $a\leq d_{H}(v)\leq b$ for each $v\in V(G)$. In this paper, we provide spectral conditions for the existence of an odd $[1,b]$-factor in a connected graph with minimum degree $\delta$ and the existence of an $[a,b]$-factor in a graph, respectively. Our results generalize and improve some previous results on perfect matchings of graphs. For $a=1$, we extend the result of O\cite{S.O} to obtain an odd $[1,b]$-factor and further improve the result of Liu, Liu and Feng\cite{W.L} for $a=b=1$. For $n\geq 3a+b-1$, we confirm the conjecture of Cho, Hyun, O and Park\cite{E.C}. We conclude some open problems in the end., Comment: 14 pages

Let $k$ and $n$ be two integers, with $k\geq 3$, $n\equiv 0\pmod k$, and $n$ sufficiently large. We determine the $(k-1)$-degree threshold for the existence of a rainbow perfect matchings in $n$-vertex $k$-uniform hypergraph. This implies the result of R\"odl, Ruci\'nski, and Szemer\'edi on the $(k-1)$-degree threshold for the existence of perfect matchings in $n$-vertex $k$-uniform hypergraphs. In our proof, we identify the extremal configurations of closeness, and consider whether or not the hypergraph is close to the extremal configuration. In addition, we also develop a novel absorbing device and generalize the absorbing lemma of R\"odl, Ruci\'nski, and Szemer\'edi.

Let $n$ be a sufficiently large integer with $n\equiv 0\pmod 4$ and let $F_i \subseteq{[n]\choose 4}$ where $i\in [n/4]$. We show that if each vertex of $F_i$ is contained in more than ${n-1\choose 3}-{3n/4\choose 3}$ edges, then $\{F_1, \ldots ,F_{n/4}\}$ admits a rainbow matching, i.e., a set of $n/4$ edges consisting of one edge from each $F_i$. This generalizes a deep result of Khan on perfect matchings in 4-uniform hypergraphs., Comment: arXiv admin note: text overlap with arXiv:2004.12561

R\"odl, Ruci\'nski, and Szemer\'edi determined the minimum $(k-1)$-degree threshold for the existence of fractional perfect matchings in $k$-uniform hypergrahs, and K\"uhn, 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 $\delta_d(H)>{n-d\choose k-d} -{n-d-(\lceil n/k\rceil-1)\choose k-d}$ 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

Let $n,s,k$ be three positive integers such that $1\leq s\leq(n-k+1)/k$ and let $[n]=\{1,\ldots,n\}$. Let $H$ be a $k$-graph with vertex set $\{1,\ldots,n\}$, and let $e(H)$ denote the number of edges of $H$. Let $\nu(H)$ and $\tau(H)$ denote the size of a largest matching and the size of a minimum vertex cover in $H$, respectively. Define $A^k_i(n,s):=\{e\in\binom{[n]}{k}:|e\cap[(s+1)i-1]|\geq i\}$ for $2\leq i\leq k$ and $HM^k_{n,s}:=\big\{e\in\binom{[n]}{k}:e\cap[s-1]\neq\emptyset\big\} \cup\big\{S\big\}\cup \big\{e\in\binom{[n]}{k}: s\in e, e\cap S\neq \emptyset\}$, where $S=\{s+1,\ldots,s+k\}$. Frankl and Kupavskii conjectured that if $\nu(H)\leq s$ and $\tau(H)>s$, then $e(H)\leq \max\{|A^k_2(n,s)|,\ldots ,|A^k_k(n,s)|,|HM^k_{n,s}|\}$. In this paper, we prove this conjecture for $k=3$ and sufficiently large $n$.

Aharoni and Howard, and, independently, Huang, Loh, and Sudakov proposed the following rainbow version of Erd\H{o}s matching conjecture: For positive integers $n,k,m$ with $n\ge km$, if each of the families $F_1,\ldots, F_m\subseteq {[n]\choose k}$ has size more than $\max\{\binom{n}{k} - \binom{n-m+1}{k}, \binom{km-1}{k}\}$, then there exist pairwise disjoint subsets $e_1,\dots, e_m$ such that $e_i\in F_i$ for all $i\in [m]$. We prove that there exists an absolute constant $n_0$ such that this rainbow version holds for $k=3$ and $n\geq n_0$. We convert this rainbow matching problem to a matching problem on a special hypergraph $H$. We then combine several existing techniques on matchings in uniform hypergraphs: find an absorbing matching $M$ in $H$; use a randomization process of Alon et al. to find an almost regular subgraph of $H-V(M)$; and find an almost perfect matching in $H-V(M)$. To complete the process, we also need to prove a new result on matchings in 3-uniform hypergraphs, which can be viewed as a stability version of a result of {\L}uczak and Mieczkowska and might be of independent interest., Comment: added two references [2,7], accepted for publication in SCIENCE CHINA Mathematics

A graph $G$ has the \emph{strong parity property} if for every subset $X\subseteq V$ with $|X|$ even, $G$ has a spanning subgraph $F$ with minimum degree at least one such that $d_F(v)\equiv 1\pmod 2$ for all $v\in X$, $d_F(y)\equiv 0\pmod 2$ for all $y\in V(G)-X$. Bujt\'as, Jendrol and Tuza (On specific factors in graphs, \emph{Graphs and Combin.}, 36 (2020), 1391-1399.) introduced the concept and conjectured that every 2-edge-connected graph with minimum degree at least three has the strong parity property. In this paper, we give a characterization for graphs to have the strong parity property and construct a counterexample to disprove the conjecture proposed by Bujt\'as, Jendrol and Tuza.

Let $a$ and $b$ be positive integers such that $a\leq b$ and $a\equiv b\pmod 2$. We say that $G$ has all $(a, b)$-parity factors if $G$ has an $h$-factor for every function $h: V(G) \rightarrow \{a,a+2,\ldots,b-2,b\}$ with $b|V(G)|$ even and $h(v)\equiv b\pmod 2$ for all $v\in V(G)$. In this paper, we prove that every graph $G$ with $n\geq 3(b+1)(a+b)$ vertices has all $(a,b)$-parity factors if $\delta(G)\geq (b^2-b)/a$, and for any two nonadjacent vertices $u,v \in V(G)$, $\max\{d_G(u),d_G(v)\}\geq \frac{bn}{a+b}$. Moreover, we show that this result is best possible in some sense.

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

K\"{u}hn, 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 $\delta_1(H)>{n-1\choose 2}-{2n/3\choose 2}$, 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 $\delta_1(F_i)>{n-1\choose 2}-{2n/3\choose 2}$ 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.

For a $k$-uniform hypergraph $H$, let $\delta_1(H)$ denote the minimum vertex degree of $H$, and $\nu(H)$ denote the size of the largest matching in $H$. In this paper, we show that for any $k\geq 3$ and $\beta>0$, there exists an integer $n_0(\beta,k)$ such that for positive integers $n\geq n_0$ and $m\leq (\frac{k}{2(k-1)}-\beta)\frac{n}{k}$, if $H$ is an $n$-vertex $k$-graph with $\delta_1(H)>{{n-1}\choose {k-1}}-{{n-m}\choose {k-1}},$ then $\nu(H)\geq m$. This improves upon earlier results of Bollob\'{a}s, Daykin and Erd\H{o}s (1976) for the range $n> 2k^3(m+1)$ and Huang and Zhao (2017) for the range $n\geq 3k^2 m$.

Let $r\geq 3$ be an integer and $G$ be a graph. Let $\delta(G), \Delta(G)$, $\alpha(G)$ and $\mu(G)$ denotes minimum degree, maximum degree, independence number and matching number of $G$, respectively. Recently, Caro, Davila and Pepper proved $\delta(G)\alpha(G)\leq \Delta(G)\mu(G)$. Mohr and Rautenbach characterized the extremal graphs for non-regular graphs and 3-regular graphs. In this note, we characterize the extremal graphs for all $r$-regular graphs in term of Gallai-Edmonds Structure Theorem, which extends Mohr and Rautenbach's result.

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 $\delta_l(H)$ is best possible. When $k=3$, with the help of an absorbing lemma of H\'{a}n, 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.