arXiv subjects

Xiaozheng Chen

Publications and source records attributed to Xiaozheng Chen.

8 recordsLinked to original sources

arXiv2026

Statistical Depth for Interval Data: A Closeness-Based Approach and Its Applications

With the rapid development of technology, interval data have been increasingly widely used in various fields, as they can effectively carry information on measurement uncertainty and original variability. Reasonable ranking and outlier screening of interval data are crucial for exploring their value, and statistical depth theory is an important tool to achieve this goal. Existing statistical depth methods are mostly defined for ordinary point-valued data, which are difficult to adapt to the characteristics of interval data and cannot meet the needs of their ranking and outlier detection. Therefore, this paper proposes a statistical depth suitable for interval data and conducts relevant application verification.

stat.ME↗

arXiv2026

On Erdős Problem 767: Cycles with Chords

For integers $k\ge 1$ and $n\ge k+2$, let $g_k(n)$ be the maximum number of edges in an $n$-vertex graph containing no cycle with a vertex incident with at least $k$ chords. Erdős conjectured that $g_k(n)=(k+1)(n-k-1)$ for $n\ge 2k+2$. Lewin found a counterexample. Bollobás later conjectured that there exists a function $n(k)$ such that $g_k(n)=(k+1)(n-k-1)$ for all $n\ge n(k)$. Jiang confirmed this by proving the formula for all $n\ge3k+3$ when $k\ge1$. In this paper, we determine $g_k(n)$ completely. For all $k\ge1$ and $n\ge k+2$, we prove $g_k(n)=\big\{\lfloor\frac{(k+1)n}{2}\rfloor,\max\{a(n-a)+\lfloor\frac{a(k+1-a)}{2}\rfloor : a\in\mathbb Z,\; \lfloor {(k+1)}/{2}\rfloor+1\le a\le k+1\}\big\}$. For $k\ge2$, we prove $g_k(n)=(k+1)(n-k-1)$ when $n\ge \lceil(5k+1)/2\rceil$, and this threshold is sharp. Our proof builds on the method developed by Ma and the second author in [Ma and Ning, 2020].

math.CO↗

arXiv2026

Cycle lengths and chords under chromatic and degree constraints

We mainly consider three problems on cycle lengths and cycles with chords in graphs: (a) Gao, Huo, and Ma \cite[Question~1.5]{GaoHuoMa2021} asked whether, for every fixed $k\ge3$, there is a function $f_k(n)\to\infty$ such that every $n$-vertex $(k+1)$-critical graph contains $f_k(n)$ consecutive cycle lengths. (b) Let $g_k(n)$ be the maximum integer $t$ such that every $n$-vertex $k$-critical graph with $k\ge4$ contains an odd cycle with at least $t$ chords. Voss conjectured (see \cite[pp.~168]{VossBook}) that $g_k(n)\to\infty$ as $n\to\infty$ for each $k\ge4$, which extends a 1976 conjecture of Erdős (see also Erdős Problem~1091 \cite{Bloom1091}). (c) Kára and Král \cite{KaraKral2003} conjectured that every graph on $31$ vertices with minimum degree at least $8$ contains a cycle with at least $31$ chords. We answer question (a) in the negative for $k=3$, and disprove conjecture (b) for all $k\ge5$. We point out the work of Alexeev-Putterman-Sawhney-Sellke-Valiant (2026) on Erdős Problem 1901 disproves the case $k=4$ for conjecture (b). We prove conjecture (c). We also discuss two other related problems in the part of concluding remark.

math.CO↗

arXiv2023

Short rainbow cycles in edge-colored graphs

A famous conjecture of Caccetta and Häggkvist (CHC) states that a directed graph $D$ with $n$ vertices and minimum outdegree at least $r$ has a directed cycle of length at most $\lceil \frac{n}{r}\rceil$. In 2017, Aharoni proposed the following generalization: an edge-colored graph $G$ with $n$ vertices, $n$ color classes of size at least $r$ has a rainbow cycle of length at most $\lceil \frac{n}{r}\rceil$. Since CHC can be seen as the case of Aharoni's Conjecture: color classes in the color partition are monochromatic stars centered at distinct vertices, one way to study Aharoni's Conjecture is to structure the color classes as each color class is either a star, a triangle or contains a matching of size 2. Guo improved the upper bound in Aharoni's Conjecture to $O(\log n)$ in some mixed cases when the color classes are not necessarily stars. In this paper, we extend Guo's results. Our main result is as follows: Let $G$ an edge-colored graph on $n$ vertices and $n$ color classes, if at least $αn$ color classes are either a matching of size 2 or a triangle for $α>\frac{1}{2}$, then $G$ contains a rainbow cycle of length $O(\log n)$. We also prove that the $\log n$ bound is the right order of magnitude.

math.CO↗

arXiv2023

Rainbow triangles sharing one common vertex or edge

Let $G$ be an edge-colored graph on $n$ vertices. For a vertex $v$, the \emph{color degree} of $v$ in $G$, denoted by $d^c(v)$, is the number of colors appearing on the edges incident with $v$. Denote by $δ^c(G)=\min\{d^c(v):v\in V(G)\}$. By a theorem of H. Li, an $n$-vertex edge-colored graph $G$ contains a rainbow triangle if $δ^c(G)\geq \frac{n+1}{2}$. Inspired by this result, we consider two related questions concerning edge-colored books and friendship subgraphs of edge-colored graphs. Let $k\geq 2$ be a positive integer. We prove that if $δ^c(G)\geq \frac{n+k-1}{2}$ where $n\geq 3k-2$, then $G$ contains $k$ rainbow triangles sharing one common edge; and if $δ^c(G)\geq \frac{n+2k-3}{2}$ where $n\geq 2k+9$, then $G$ contains $k$ rainbow triangles sharing one common vertex. The special case $k=2$ of both results improves H. Li's theorem. The main novelty of our proof of the first result is a combination of the recent new technique for finding rainbow cycles due to Czygrinow, Molla, Nagle, and Oursler and some recent counting technique from \cite{LNSZ}. The proof of the second result is with the aid of the machine implicitly in the work of Turán numbers for matching numbers due to Erdős and Gallai.

math.CO↗

arXiv2020

Rainbow triangles in edge-colored complete graphs

Let $G$ be a graph of order $n$ with an edge-coloring $c$, and let $δ^c(G)$ denote the minimum color-degree of $G$. A subgraph $F$ of $G$ is called rainbow if any two edges of $F$ have distinct colors. There have been a lot results in the existing literature on rainbow triangles in edge-colored complete graphs. Fujita and Magnant showed that for an edge-colored complete graph $G$ of order $n$, if $δ^c(G)\geq \frac{n+1}{2}$, then every vertex of $G$ is contained in a rainbow triangle. In this paper, we show that if $δ^c(G)\geq \frac{n+k}{2}$, then every vertex of $G$ is contained in at least $k$ rainbow triangles, which can be seen as a generalization of their result. Li showed that for an edge-colored graph $G$ of order $n$, if $δ^c(G)\geq \frac{n+1}{2}$, then $G$ contains a rainbow triangle. We show that if $G$ is complete and $δ^c(G)\geq \frac{n}{2}$, then $G$ contains a rainbow triangle and the bound is sharp. Hu et al. showed that for an edge-colored graph $G$ of order $n\geq 20$, if $δ^c(G)\geq \frac{n+2}{2}$, then $G$ contains two vertex-disjoint rainbow triangles. We show that if $G$ is complete with order $n\geq 8$ and $δ^c(G)\geq \frac{n+1}{2}$, then $G$ contains two vertex-disjoint rainbow triangles. Moreover, we improve the result of Hu et al. from $n\geq 20$ to $n\geq 7$, the best possible.

math.CO↗

arXiv2020

Note on rainbow cycles in edge-colored graphs

Let $G$ be a graph of order $n$ with an edge-coloring $c$, and let $δ^c(G)$ denote the minimum color degree of $G$. A subgraph $F$ of $G$ is called rainbow if all edges of $F$ have pairwise distinct colors. There have been a lot results on rainbow cycles of edge-colored graphs. In this paper, we show that (i) if $δ^c(G)>\frac{3n-3}{4}$, then every vertex of $G$ is contained in a rainbow triangle; (ii) $δ^c(G)>\frac{3n}{4}$, then every vertex of $G$ is contained in a rainbow $C_4$; and (iii) if $G$ is complete, $n\geq 8k-18$ and $δ^c(G)>\frac{n-1}{2}+k$, then $G$ contains a rainbow cycle of length at least $k$. Some gaps in previous publications are also found and corrected.

math.CO↗

arXiv2020

Proper vertex-pancyclicity of edge-colored complete graphs without joint monochromatic triangles

In an edge-colored graph $(G,c)$, let $d^c(v)$ denote the number of colors on the edges incident with a vertex $v$ of $G$ and $δ^c(G)$ denote the minimum value of $d^c(v)$ over all vertices $v\in V(G)$. A cycle of $(G,c)$ is called proper if any two adjacent edges of the cycle have distinct colors. An edge-colored graph $(G,c)$ on $n\geq 3$ vertices is called properly vertex-pancyclic if each vertex of $(G,c)$ is contained in a proper cycle of length $\ell$ for every $\ell$ with $3 \le \ell \le n$. Fujita and Magnant conjectured that every edge-colored complete graph on $n\geq 3$ vertices with $δ^c(G)\geq \frac{n+1}{2}$ is properly vertex-pancyclic. Chen, Huang and Yuan partially solve this conjecture by adding an extra condition that $(G,c)$ does not contain any monochromatic triangle. In this paper, we show that this conjecture is true if the edge-colored complete graph contain no joint monochromatic triangles.

math.CO↗