arXiv subjects

Chuandong Xu

Publications and source records attributed to Chuandong Xu.

10 recordsLinked to original sources

arXiv2026

A finite forbidden family with superlinear surplus and non-join extremal graphs

We give a common counterexample to two product-structure conjectures in extremal graph theory. More precisely, we construct a fixed nonempty finite family $\mathcal L$ with $p(\mathcal L)=2$ such that, for some $c>0$, \[ \operatorname{ex}(n,\mathcal L)>t_2(n)+cn^{3/2} \] for every sufficiently large $n$. Nevertheless, at every such order there is an $\mathcal L$-extremal graph whose complement is connected and which therefore admits no decomposition as a join of two nonempty graphs. This superlinear surplus also forces the decomposition family of $\mathcal L$ to contain no forest. The construction uses endpoint-injective repair with a finite obstruction family admitting an exact extremal formula and equality classification.

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arXiv2026

A finite forbidden family with superlinear surplus and no three-factor product extremizers

We construct a fixed finite family $\mathcal L$ of ordinary forbidden subgraphs with $p(\mathcal L)=3$ and a constant $c>0$ such that $$ex(n,\mathcal L)>t_3(n)+cn^{3/2}$$ at every sufficiently large order. Nevertheless, the complement of every sufficiently large $\mathcal L$-extremal graph has at most two connected components. In particular, no such extremal graph is a complete join of three graphs of positive order. This gives a negative answer to a natural existence-only question motivated by the Simonovits Product Conjecture, in which one asks only for one product extremizer at each sufficiently large order.

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arXiv2025

Sphere intersections and incidences over finite fields

We bound the number of incidences between points and spheres in finite vector spaces by bounding the sum of the number of points in the pairwise intersections of the spheres. We obtain new incidence bounds that are interesting when the number of spheres is not too large. Our approach also leads to an elementary proof of the Iosevich-Rudnev bound on the Erdős-Falconer distance problem in odd dimensions.

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arXiv2025

Embedding edge-colored graphs in expanders with roll-back

We introduce a method to embed edge-colored graphs into families of expander graphs, which generalizes a framework developed by Draganić, Krivelevich, and Nenadov (2022). As an application, we show that each family of sufficiently pseudo-random graphs on $n$ vertices contains every edge-colored subdivision of $K_Δ$, provided that the distance between branch vertices in the subdivision is large enough, the average degree of each graph in the family is at least $(1+o(1))Δ$, and the number of vertices in the subdivision is at most $(1-o(1))n$. This work is motivated in part by the problem of finding structures in distance graphs defined over finite vector spaces. For $d\ge 2$ and an odd prime power $q$, consider the vector space $\mathbb{F}_q^d$ over the finite field $\mathbb{F}_q$, where the distance between two points $(x_1,\ldots,x_d)$ and $(y_1,\ldots,y_d)$ is defined to be $\sum_{i=1}^d (x_i-y_i)^2$. A distance graph is a graph associated with a non-zero distance to each of its edges. We show that large subsets of vector spaces over finite fields contain every distance graph that is a nearly spanning subdivision of a complete graph, provided that the distance between branching vertices in the subdivision is large enough.

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arXiv2024

Cooperative coloring of some graph families

In a family ${G_1, G_2, \ldots, G_m}$ of graphs sharing the same vertex set $V$, a cooperative coloring involves selecting one independent set $I_i$ from $G_i$ for each $i\in \{1,2,\ldots,m\}$ such that $\bigcup_{i=1}^m I_i = V$. For a graph class $\mathcal{G}$, let $m_{\mathcal{G}}(d)$ denote the minimum $m$ required to ensure that any graph family ${G_1, G_2, \ldots, G_m}$ on the same vertex set, where $G_i\in\mathcal{G}$ and $Δ(G_i)\leq d$ for each $i\in \{1,2,\ldots,m\}$, admits a cooperative coloring. For the graph classes $\mathcal{T}$ (trees) and $\mathcal{W}$ (wheels), we find that $m_\mathcal{T}(3)=4$ and $m_\mathcal{W}(4)=5$. Also, we prove that $m_{\mathcal{B}^*}(d)=O(\log_2 d)$ and $m_{\mathcal{L}}(d)=O\left(\frac{\log d}{\log\log d}\right)$, where $\mathcal{B}^*$ represents the class of graphs whose components are balanced complete bipartite graphs, and $\mathcal{L}$ represents the class of graphs whose components are generalized theta graphs.

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arXiv2020

On characterizing the critical graphs for matching Ramsey numbers

Given simple graphs $H_{1},H_{2},\ldots,H_{c}$, the Ramsey number $r(H_{1},H_{2},\ldots,H_{c})$ is the smallest positive integer $n$ such that every edge-colored $K_{n}$ with $c$ colors contains a subgraph in color $i$ isomorphic to $H_{i}$ for some $i\in\{1,2,\ldots,c\}$. The critical graphs for $r(H_1,H_2,\ldots,H_c)$ are edge-colored complete graphs on $r(H_1,H_2,\ldots,H_c)-1$ vertices with $c$ colors which contain no subgraphs in color $i$ isomorphic to $H_{i}$ for any $i\in \{1,2,\ldots,c\}$. For $n_1\geq n_2\geq \ldots\geq n_c\geq 1$, Cockayne and Lorimer (The Ramsey number for stripes, {\it J.\ Austral.\ Math.\ Soc.} \textbf{19} (1975), 252--256.) showed that $r(n_{1}K_{2},n_{2}K_{2},\ldots,n_{c}K_{2})=n_{1}+1+ \sum\limits_{i=1}^c(n_{i}-1)$, in which $n_{i}K_{2}$ is a matching of size $n_{i}$. Using the Gallai-Edmonds Theorem, we characterized all the critical graphs for $r(n_{1}K_{2},n_{2}K_{2},\ldots,n_{c}K_{2})$, implying a new proof for this Ramsey number.

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arXiv2019

Properly colored $C_{4}$'s in edge-colored graphs

When many colors appear in edge-colored graphs, it is only natural to expect rainbow subgraphs to appear. This anti-Ramsey problem has been studied thoroughly and yet there remain many gaps in the literature. Expanding upon classical and recent results forcing rainbow triangles to appear, we consider similar conditions which force the existence of a properly colored copy of $C_{4}$.

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arXiv2019

On sufficient conditions for rainbow cycles in edge-colored graphs

Let $G$ be an edge-colored graph. We use $e(G)$ and $c(G)$ to denote the number of edges of $G$ and the number of colors appearing on $E(G)$, respectively. For a vertex $v\in V(G)$, the \emph{color neighborhood} of $v$ is defined as the set of colors assigned to the edges incident to $v$. A subgraph of $G$ is \emph{rainbow} if all of its edges are assigned with distinct colors. The well-known Mantel's theorem states that a graph $G$ on $n$ vertices contains a triangle if $e(G)\geq\lfloor\frac{n^2}{4}\rfloor+1$. Rademacher (1941) showed that $G$ contains at least $\lfloor\frac{n}{2}\rfloor$ triangles under the same condition. Li, Ning, Xu and Zhang (2014) proved a rainbow version of Mantel's theorem: An edge-colored graph $G$ has a rainbow triangle if $e(G)+c(G)\geq n(n+1)/2$. In this paper, we first characterize all graphs $G$ satisfying $e(G)+c(G)\geq n(n+1)/2-1$ but containing no rainbow triangles. Motivated by Rademacher's theorem, we then characterize all graphs $G$ which satisfy $e(G)+c(G)\geq n(n+1)/2$ but contain only one rainbow triangle. We further obtain two results on color neighborhood conditions for the existence of rainbow short cycles. Our results improve a previous theorem due to Broersma, Li, Woeginger, and Zhang (2005). Moreover, we provide a sufficient condition in terms of color neighborhood for the existence of a specified number of vertex-disjoint rainbow cycles.

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arXiv2013

Rainbow triangles in edge-colored graphs

Let $G$ be an edge-colored graph. The color degree of a vertex $v$ of $G$, is defined as the number of colors of the edges incident to $v$. The color number of $G$ is defined as the number of colors of the edges in $G$. A rainbow triangle is one in which every pair of edges have distinct colors. In this paper we give some sufficient conditions for the existence of rainbow triangles in edge-colored graphs in terms of color degree, color number and edge number. As a corollary, a conjecture proposed by Li and Wang (Color degree and heterochromatic cycles in edge-colored graphs, European J. Combin. 33 (2012) 1958--1964) is confirmed.

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arXiv2011

Extreme tenacity of graphs with given order and size

Computer or communication networks are so designed that they do not easily get disrupted under external attack and, moreover, these are easily reconstructible if they do get disrupted. These desirable properties of networks can be measured by various graph parameters, such as connectivity, toughness, scattering number, integrity, tenacity, rupture degree and edge-analogues of some of them. Among these parameters, the tenacity and rupture degree are two better ones to measure the stability of a network. In this paper we consider two extremal problems on the tenacity of graphs: Determine the minimum and maximum tenacity of graphs with given order and size. We give a complete solution to the first problem, while for the second one, it turns out that the problem is much more complicated than that of the minimum case. We determine the maximum tenacity of trees and unicyclic graphs with given order and show the corresponding extremal graphs. These results are helpful in constructing stable networks with lower costs. The paper concludes with a discussion of a related problem on the edge vulnerability parameters of graphs.

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