arXiv subjects
Publications and source records attributed to Zhong Huang.
In this note, we prove a theorem covering Chartrand, Kaigars, and Lick's theorem in [Proc. Amer. Math. Soc. 32 (1972), 63-68]. As an application, we give a simpler proof of theorem proved by Mader [J. Graph Theory 65 (2010), 61-69. (Theorem 1)].
Let $p,q$ be two integers with $p\geq q$. Given a finite graph $F$ with no isolated vertices, the generalized Ramsey achievement game of $F$ on the complete graph $K_n$, denoted by $(p,q;K_n,F,+)$, is played by two players called Alice and Bob. In each round, Alice firstly chooses $p$ uncolored edges $e_1,e_2,...,e_p$ and colors it blue, then Bob chooses $q$ uncolored edge $f_1,f_2,...,f_q$ and colors it red; the player who can first complete the formation of $F$ in his (or her) color is the winner. The generalized achievement number of $F$, denoted by ${a}(p,q;F)$ is defined to be the smallest $n$ for which Alice has a winning strategy. If $p=q=1$, then it is denoted by ${a}(F)$, which is the classical achievement number of $F$ introduced by Harary in 1982. If Alice aims to form a blue $F$, and the goal of Bob is to try to stop him, this kind of game is called the first player game by Bollobás. Let ${a}^*(F)$ be the smallest positive integer $n$ for which Alice has a winning strategy in the first player game. A conjecture due to Harary states that the minimum value of ${a}(T)$ is realized when $T$ is a path and the maximum value of ${a}(T)$ is realized when $T$ is a star among all trees $T$ of order $n$. He also asked which graphs $F$ satisfy $a^*(F)=a(F)$? In this paper, we proved that $n\leq {a}(p,q;T)\leq n+q\left\lfloor (n-2)/p \right\rfloor$ for all trees $T$ of order $n$, and obtained a lower bound of ${a}(p,q;K_{1,n-1})$, where $K_{1,n-1}$ is a star. We proved that the minimum value of ${a}(T)$ is realized when $T$ is a path which gives a positive solution to the first part of Harary's conjecture, and ${a}(T)\leq 2n-2$ for all trees of order $n$. We also proved that for $n\geq 3$, we have $2n-2-\sqrt{(4n-8)\ln (4n-4)}\leq a(K_{1,n-1})\leq 2n-2$ with the help of a theorem of Alon, Krivelevich, Spencer and Szabó. We proved that $a^*(P_n)=a(P_n)$ for a path $P_n$.
We identified a new ultra-distant comet C/2019 E3 (ATLAS) exhibiting preperihelion cometary activity at heliocentric distances $\gtrsim\!20$ au, making it the fourth member of this population after C/2010 U3 (Boattini), C/2014 UN$_{271}$ (Bernardinelli-Bernstein), and C/2017 K2 (PANSTARRS). From serendipitous archival data, we conducted analyses of the comet, finding that the activity was consistent with steady-state behaviour, suggestive of sublimation of supervolatiles, that the cross-section of dust increased gradually on the inbound leg of the orbit, varying with heliocentric distances as $r_{\rm H}^{-1.5 \pm 0.4}$, and that the dust was produced at a rate of $\gtrsim\!10^2$ kg s$^{-1}$ within the observed timespan. Our modelling of the largely symmetric morphology of the comet suggests that the dust environment was likely dominated by mm-scale dust grains ejected at speeds $\lesssim\!0.4$ m s$^{-1}$ from the sunlit hemisphere of the nucleus. Assuming a typical geometric albedo of 0.05 and adopting several simplistic thermophysical models, we estimated the nucleus to be at least $\sim\!3$ km across. We also measured the colour of the comet to be consistent with other long-period comets, except being slightly bluer in $g-r$. With our astrometric measurements, we determined an improved orbit of the comet, based upon which we derived that the comet is dynamically new and that its perihelion distance will further shrink due to the Galactic tide. We conclude the paper by comparing the known characteristics of the known ultra-distant comets.
In 1982, Harary introduced the concept of Ramsey achievement game on graphs. Given a graph $F$ with no isolated vertices. Consider the following game played on the complete graph $K_n$ by two players Alice and Bob. First, Alice colors one of the edges of $K_n$ blue, then Bob colors a different edge red, and so on. The first player who can complete the formation of $F$ in his color is the winner. The minimum $n$ for which Alice has a winning strategy is the achievement number of $F$, denoted by $a(F)$. If we replace $K_n$ in the game by the completed bipartite graph $K_{n,n}$, we get the bipartite achievement number, denoted by $\operatorname{ba}(F)$. In his seminal paper, Harary proposed an open problem of determining bipartite achievement numbers for trees. In this paper, we correct $\operatorname{ba}(mK_2)=m+1$ to $m$ and disprove $\operatorname{ba}(K_{1,m})=2m-2$ from Erickson and Harary, and extend their results on bipartite achievement numbers. We also find the exact values of achievement numbers for matchings, and the exact values or upper and lower bounds of bipartite achievement numbers on matchings, stars, and double stars. Our upper bounds are obtained by deriving efficient winning strategies for Alice.
The recently proposed Graph Convolutional Networks (GCNs) have achieved significantly superior performance on various graph-related tasks, such as node classification and recommendation. However, currently researches on GCN models usually recursively aggregate the information from all the neighbors or randomly sampled neighbor subsets, without explicitly identifying whether the aggregated neighbors provide useful information during the graph convolution. In this paper, we theoretically analyze the affection of the neighbor quality over GCN models' performance and propose the Neighbor Enhanced Graph Convolutional Network (NEGCN) framework to boost the performance of existing GCN models. Our contribution is three-fold. First, we at the first time propose the concept of neighbor quality for both node classification and recommendation tasks in a general theoretical framework. Specifically, for node classification, we propose three propositions to theoretically analyze how the neighbor quality affects the node classification performance of GCN models. Second, based on the three proposed propositions, we introduce the graph refinement process including specially designed neighbor evaluation methods to increase the neighbor quality so as to boost both the node classification and recommendation tasks. Third, we conduct extensive node classification and recommendation experiments on several benchmark datasets. The experimental results verify that our proposed NEGCN framework can significantly enhance the performance for various typical GCN models on both node classification and recommendation tasks.
An edge-cut $R$ of an edge-colored connected graph is called a rainbow-cut if no two edges in the edge-cut are colored the same. An edge-colored graph is rainbow disconnected if for any two distinct vertices $u$ and $v$ of the graph, there exists a $u$-$v$-rainbow-cut separating them. For a connected graph $G$, the rainbow disconnection number of $G$, denoted by rd$(G)$, is defined as the smallest number of colors that are needed in order to make $G$ rainbow disconnected. In this paper, we first give some tight upper bounds for rd$(G)$, and moreover, we completely characterize the graphs which meet the upper bound of the Nordhaus-Gaddum type results obtained early by us. Secondly, we propose a conjecture that $λ^+(G)\leq \textnormal{rd}(G)\leq λ^+(G)+1$, where $λ^+(G)$ is the upper edge-connectivity, and prove the conjecture for many classes of graphs, to support it. Finally, we give the relationship between rd$(G)$ of a graph $G$ and the rainbow vertex-disconnection number rvd$(L(G))$ of the line graph $L(G)$ of $G$.
The concept of rainbow connection number of a graph was introduced by Chartrand et al. in 2008. Inspired by this concept, other concepts on colored version of connectivity in graphs were introduced, such as the monochromatic connection number by Caro and Yuster in 2011, the proper connection number by Borozan et al. in 2012, and the conflict-free connection number by Czap et al. in 2018, as well as some other variants of connection numbers later on. Chakraborty et al. proved that to compute the rainbow connection number of a graph is NP-hard. For a long time, it has been tried to fix the computational complexity for the monochromatic connection number, the proper connection number and the conflict-free connection number of a graph. However, it has not been solved yet. Only the complexity results for the strong version, i.e., the strong proper connection number and the strong conflict-free connection number, of these connection numbers were determined to be NP-hard. In this paper, we prove that to compute each of the monochromatic connection number, the proper connection number and the conflict free connection number for a graph is NP-hard. This solves a long standing problem in this field, asked in many talks of workshops and papers.
Let $G$ be a nontrivial connected, edge-colored graph. An edge-cut $S$ of $G$ is called a rainbow cut if no two edges in $S$ are colored with a same color. An edge-coloring of $G$ is a rainbow disconnection coloring if for every two distinct vertices $s$ and $t$ of $G$, there exists a rainbow cut $S$ in $G$ such that $s$ and $t$ belong to different components of $G\setminus S$. For a connected graph $G$, the {\it rainbow disconnection number} of $G$, denoted by $rd(G)$, is defined as the smallest number of colors such that $G$ has a rainbow disconnection coloring by using this number of colors. In this paper, we show that for a connected graph $G$, computing $rd(G)$ is NP-hard. In particular, it is already NP-complete to decide if $rd(G)=3$ for a connected cubic graph. Moreover, we prove that for a given edge-colored (with an unbounded number of colors) connected graph $G$ it is NP-complete to decide whether $G$ is rainbow disconnected.
Exceptional points (EPs) are degeneracies in open wave systems with coalescence of at least two energy levels and their corresponding eigenstates. In higher dimensions, more complex EP physics not found in two-state systems is observed. We consider the emergence and interaction of multiple EPs in a four coupled optical waveguides system by non-Hermitian coupling showing a unique EP formation pattern in a phase diagram. In addition, absolute phase rigidities are computed to show the mixing of the different states in definite parameter regimes. Our results could be potentially important for developing further understanding of EP physics in higher dimensions via generalized paradigm of nonHermitian coupling for a new generation of parity-time (PT) devices.
An edge-colored graph $G$ is \emph{conflict-free connected} if any two of its vertices are connected by a path, which contains a color used on exactly one of its edges. The \emph{conflict-free connection number} of a connected graph $G$, denoted by $cfc(G)$, is the smallest number of colors needed in order to make $G$ conflict-free connected. For a graph $G,$ let $C(G)$ be the subgraph of $G$ induced by its set of cut-edges. In this paper, we first show that, if $G$ is a connected non-complete graph $G$ of order $n\geq 9$ with $C(G)$ being a linear forest and with the minimum degree %$δ(G)\geq 2$, then $cfc(G)=2$ for $4 \leq n\leq 8 $; if $δ(G)\geq \max\{3, \frac{n-4}{5}\}$, then $cfc(G)=2$. The bound on the minimum degree is best possible. Next, we prove that, if $G$ is a connected non-complete graph of order $n\geq 33$ with $C(G)$ being a linear forest and with $d(x)+d(y)\geq \frac{2n-9}{5}$ for each pair of two nonadjacent vertices $x, y$ of $V(G)$, then $cfc(G)=2$. Both bounds, on the order $n$ and the degree sum, are tight. Moreover, we prove several results concerning relations between degree conditions on $G$ and the number of cut edges in $G$.
An edge-colored graph $G$ is \emph{conflict-free connected} if, between each pair of distinct vertices, there exists a path containing a color used on exactly one of its edges. The \emph{conflict-free connection number} of a connected graph $G$, denoted by $cfc(G)$, is defined as the smallest number of colors that are needed in order to make $G$ conflict-free connected. In this paper, we determine all trees $T$ of order $n$ for which $cfc(T)=n-t$, where $t\geq 1$ and $n\geq 2t+2 $. Then we prove that $1\leq cfc(G)\leq n-1$ for a connected graph $G$, and characterize the graphs $G$ with $cfc(G)=1,n-4,n-3,n-2,n-1$, respectively. Finally, we get the Nordhaus-Gaddum-type theorem for the conflict-free connection number of graphs, and prove that if $G$ and $\overline{G}$ are connected, then $4\leq cfc(G)+cfc(\overline{G})\leq n$ and $4\leq cfc(G)\cdot cfc(\overline{G})\leq2(n-2)$, and moreover, $cfc(G)+cfc(\overline{G})=n$ or $cfc(G)\cdot cfc(\overline{G})=2(n-2)$ if and only if one of $G$ and $\overline{G}$ is a tree with maximum degree $n-2$ or a $P_5$, and the lower bounds are sharp.
An edge-colored graph $G$ is $k$-color connected if, between each pair of vertices, there exists a path using at least $k$ different colors. The $k$-color connection number of $G$, denoted by $cc_{k}(G)$, is the minimum number of colors needed to color the edges of $G$ so that $G$ is $k$-color connected. First, we prove that let $H$ be a subdivision of a connected graph $G$, then $cc_{k}(H)\leq cc_{k}(G)$. Second, we give sufficient conditions to guarantee that $cc_{k}(G)=k$ in terms of minimum degree and the number of edges for 2-connected graphs. As a byproduct, we show that almost all graphs have the $k$-color connection number $k$. At last, we investigate the relationship between the $k$-color connection number and the rainbow connection number for a connected graph. In addition, we give exact values of $k$-color connection numbers for some graph classes: subdivisions of the wheel and the complete graph, and the generalised $θ$-graph.
A path $P$ in an edge-colored graph $G$ is a \emph{proper path} if no two adjacent edges of $P$ are colored with the same color. The graph $G$ is \emph{proper connected} if, between every pair of vertices, there exists a proper path in $G$. The \emph{proper connection number} $pc(G)$ of a connected graph $G$ is defined as the minimum number of colors to make $G$ proper connected. In this paper, we study the degree sum condition for a general graph or a bipartite graph to have proper connection number 2. First, we show that if $G$ is a connected noncomplete graph of order $n\geq 5$ such that $d(x)+d(y)\geq \frac{n}{2}$ for every pair of nonadjacent vertices $x,y\in V(G)$, then $pc(G)=2$ except for three small graphs on 6, 7 and 8 vertices. In addition, we obtain that if $G$ is a connected bipartite graph of order $n\geq 4$ such that $d(x)+d(y)\geq \frac{n+6}{4}$ for every pair of nonadjacent vertices $x,y\in V(G)$, then $pc(G)=2$. Examples are given to show that the above conditions are best possible.
The progress in exploiting new electronic materials and devices has been a major driving force in solid-state physics. As a new state of matter, a Weyl semimetal (WSM), particularly a type-II WSM, hosts Weyl fermions as emergent quasiparticles and may harbor novel electrical transport properties because of the exotic Fermi surface. Nevertheless, such a type-II WSM material has not been experimentally observed in nature. In this work, by performing systematic magneto-transport studies on thin films of a predicted material candidate WTe2, we observe notable angle-sensitive (between the electric and magnetic fields) negative longitudinal magnetoresistance (MR), which can likely be attributed to the chiral anomaly in WSM. This phenomenon also exhibits strong planar orientation dependence with the absence of negative longitudinal MR along the tungsten chains (a axis), which is consistent with the distinctive feature of a type-II WSM. By applying a gate voltage, we demonstrate that the Fermi energy can be tuned through the Weyl points via the electric field effect; this is the first report of controlling the unique transport properties in situ in a WSM system. Our results have important implications for investigating simulated quantum field theory in solid-state systems and may open opportunities for implementing new types of electronic applications, such as field-effect chiral electronic devices.