arXiv subjects
Publications and source records attributed to Xiaoyu He.
In 1971, Tomescu conjectured that every connected graph $G$ on $n$ vertices with chromatic number $k\geq4$ has at most $k!(k-1)^{n-k}$ proper $k$-colorings. Recently, Knox and Mohar proved Tomescu's conjecture for $k=4$ and $k=5$. In this paper, we complete the proof of Tomescu's conjecture for all $k\ge 4$, and show that equality occurs if and only if $G$ is a $k$-clique with trees attached to each vertex.
The weight decision problem, which requires to determine the Hamming weight of a given binary string, is a natural and important problem, with applications in cryptanalysis, coding theory, fault-tolerant circuit design and so on. In particular, both Deutsch-Jozsa problem and Grover search problem can be interpreted as special cases of weight decision problems. In this work, we investigate the exact quantum query complexity of weight decision problems, where the quantum algorithm must always output the correct answer. More specifically we consider a partial Boolean function which distinguishes whether the Hamming weight of the length-$n$ input is $k$ or it is $l$. Our contribution includes both upper bounds and lower bounds for the precise number of queries. Furthermore, for most choices of $(\frac{k}{n},\frac{l}{n})$ and sufficiently large $n$, the gap between our upper and lower bounds is no more than one. To get the results, we first build the connection between Chebyshev polynomials and our problem, then determine all the boundary cases of $(\frac{k}{n},\frac{l}{n})$ with matching upper and lower bounds, and finally we generalize to other cases via a new \emph{quantum padding} technique. This quantum padding technique can be of independent interest in designing other quantum algorithms.
String matching is one of the most fundamental problems in computer science. A natural problem is to determine the number of characters that need to be queried (i.e. the decision tree complexity) in a string in order to decide whether this string contains a certain pattern. Rivest showed that for every pattern $p$, in the worst case any deterministic algorithm needs to query at least $n-|p|+1$ characters, where $n$ is the length of the string and $|p|$ is the length of the pattern. He further conjectured that this bound is tight. By using the adversary method, Tuza disproved this conjecture and showed that more than one half of binary patterns are {\em evasive}, i.e. any algorithm needs to query all the characters (see Section 1.1 for more details). In this paper, we give a query algorithm which settles the decision tree complexity of string matching except for a negligible fraction of patterns. Our algorithm shows that Tuza's criteria of evasive patterns are almost complete. Using the algebraic approach of Rivest and Vuillemin, we also give a new sufficient condition for the evasiveness of patterns, which is beyond Tuza's criteria. In addition, our result reveals an interesting connection to \emph{Skolem's Problem} in mathematics.
We introduce a novel generalization of the Clauser-Horne-Shimony-Holt (CHSH) game to a multiplayer setting, i.e., Hypercube game, where all $m$ players are required to assign values to vertices on corresponding facets of an $m$-dimensional hypercube. The players win if and only if their answers satisfy both parity and consistency conditions. We completely characterize the maximum winning probabilities (game value) under classical, quantum and no-signalling strategies, respectively. In contrast to the original CHSH game designed to demonstrate the superiority of quantumness, we find that the quantum advantages in the Hypercube game significantly decrease as the number of players increase. Notably, the quantum value decays exponentially fast to the classical value as $m$ increases, while the no-signalling value always remains to be one.
We present a simple model which allows us to explain the physical nature of the oscillating entropy. We consider an ensemble of qubits interacting with thermal two-level systems. The entropy of the qubits oscillates between zero and the value of entropy of the thermal systems. We show that the oscillations of the entropy can be clearly explained by the precession of the real or effective spins of the qubits.
When $k$ is a constant at least $3$, a sequence $S$ of positive integers is called $k$-GP-free if it contains no nontrivial $k$-term geometric progressions. Beiglbök, Bergelson, Hindman and Strauss first studied the existence of a $ $$k$-GP-free sequence with bounded gaps. In a previous paper the author gave a partial answer to this question by constructing a $6$-GP-free sequence $S$ with gaps of size $O(\exp(6\log n/\log\log n))$. We generalize this problem to allow the gap function $k$ to grow to infinity, and ask: for which pairs of functions $(h,k)$ do there exist $k$-GP-free sequences with gaps of size $O(h)$? We show that whenever $(k(n)-3)\log h(n)\log\log h(n)\ge4\log2\cdot\log n$ and $h,k$ satisfy mild growth conditions, such a sequence exists.
For a finite abelian group $G$ and a positive integer $k$, let $s_{k}(G)$ denote the smallest integer $\ell\in\mathbb{N}$ such that any sequence $S$ of elements of $G$ of length $|S|\geq\ell$ has a zero-sum subsequence with length $k$. The celebrated Erdős-Ginzburg-Ziv theorem determines $s_{n}(C_{n})=2n-1$ for cyclic groups $C_{n}$, while Reiher showed in 2007 that $s_{n}(C_{n}^{2})=4n-3$. In this paper we prove for a $p$-group $G$ with exponent $\exp(G)=q$ the upper bound $s_{kq}(G)\le(k+2d-2)q+3D(G)-3$ whenever $k\geq d$, where $d=\Big\lceil\frac{D(G)}{q}\Big\rceil$ and $p$ is a prime satisfying $p\ge2d+3\Big\lceil\frac{D(G)}{2q}\Big\rceil-3$, where $D(G)$ is the Davenport constant of the finite abelian group $G$. This is the correct order of growth in both $k$ and $d$. As a corollary, we show $s_{kq}(C_{q}^{d})=(k+d)q-d$ whenever $k\geq p+d$ and $2p\geq7d-3$, resolving a case of the conjecture of Gao, Han, Peng, and Sun that $s_{k\exp(G)}(G)=k\exp(G)+D(G)-1$ whenever $k\exp(G)\geq D(G)$. We also obtain a general bound $s_{kn}(C_{n}^{d})\leq9kn$ for $n$ with large prime factors and $k$ sufficiently large. Our methods are inspired by the algebraic method of Kubertin, who proved that $s_{kq}(C_{q}^{d})\leq(k+Cd^{2})q-d$ whenever $k\geq d$ and $q$ is a prime power.
Various authors, including McNew, Nathanson and O'Bryant, have recently studied the maximal asymptotic density of a geometric progression free sequence of positive integers. In this paper we prove the existence of geometric progression free sequences with small gaps, partially answering a question posed originally by Beiglböck et al. Using probabilistic methods we prove the existence of a sequence $T$ not containing any $6$-term geometric progressions such that for any $x\geq1$ and $\varepsilon>0$ the interval $[x,x+C_{\varepsilon}\exp((C+\varepsilon)\log x/\log\log x)]$ contains an element of $T$, where $C=\frac{5}{6}\log2$ and $C_{\varepsilon}>0$ is a constant depending on $\varepsilon$. As an intermediate result we prove a bound on sums of functions of the form $f(n)=\exp(-d_{k}(n))$ in very short intervals, where $d_{k}(n)$ is the number of positive $k$-th powers dividing $n$, using methods similar to those that Filaseta and Trifonov used to prove bounds on the gaps between $k$-th power free integers.
The cross number of a sequence over a finite abelian group $G$ is the sum of the inverse orders of the terms of that sequence. We study two group invariants, the maximal cross number of a zero-sum free sequence over $G$, called $\mathsf{k}(G)$, introduced by Krause, and the maximal cross number of a unique factorization sequence over $G$, called $K_{1}(G)$, introduced by Gao and Wang. Conjectured formulae for $\mathsf{k}(G)$ and $\mathsf{K}_{1}(G)$ are known, but only some special cases are proved for either. We show structural results about maximal cross number sequences that allow us to prove an inductive theorem giving conditions under which the conjectured values of $\mathsf{k}$ and $\mathsf{K}_{1}$ must be correct for $G\oplus C_{p^α}$ if they are correct for a group $G$. As a corollary of this result we prove the conjectured values of $\mathsf{k}(G)$ and $\mathsf{K}_{1}(G)$ for cyclic groups $C_{n}$, given that the prime factors of $n$ are far apart. Our methods also prove the $\mathsf{K}_{1}(G)$ conjecture for rank two groups of the form $C_{n}\oplus C_{q}$, where $q$ is the largest or second largest prime dividing $n$, and the prime factors of $n$ are far apart, and the $\mathsf{k}(G)$ conjecture for groups of the form $C_{n}\oplus H_{q}$, where the prime factors of $n$ are far apart, $q$ is the largest prime factor of $n$, and $H_{q}$ is an arbitrary finite abelian $q$-group. Finally, we pose a conjecture about the structure of maximal-length unique factorization sequences over elementary $p$-groups, which is a major roadblock to extending the $\mathsf{K}_{1}$ conjecture to groups of higher rank, and formulate a general question about the structure of maximal zero-sum free and unique factorization sequences with respect to arbitrary weighting functions.
The combinatorial theory of rotor-routers has connections with problems of statistical mechanics, graph theory, chaos theory, and computer science. A rotor-router network defines a deterministic walk on a digraph G in which a particle walks from a source vertex until it reaches one of several target vertices. Motivated by recent results due to Giacaglia et al., we study rotor-router networks in which all non-target vertices have the same type. A rotor type r is universal if every hitting sequence can be achieved by a homogeneous rotor-router network consisting entirely of rotors of type r. We give a conjecture that completely classifies universal rotor types. Then, this problem is simplified by a theorem we call the Reduction Theorem that allows us to consider only two-state rotors. A rotor-router network called the compressor, because it tends to shorten rotor periods, is introduced along with an associated algorithm that determines the universality of almost all rotors. New rotor classes, including boppy rotors, balanced rotors, and BURD rotors, are defined to study this algorithm rigorously. Using the compressor the universality of new rotor classes is proved, and empirical computer results are presented to support our conclusions. Prior to these results, less than 100 of the roughly 260,000 possible two-state rotor types of length up to 17 were known to be universal, while the compressor algorithm proves the universality of all but 272 of these rotor types.