arXiv subjects

Xiao-Chuan Liu

Publications and source records attributed to Xiao-Chuan Liu.

At least 19 recordsLinked to original sources

arXiv2026

Minimal sets for torus homeomorphisms with an irrational circle factor

We study minimal sets of torus homeomorphisms admitting an irrational circle factor whose fibres are thin essential annular continua. For totally irrational pseudo-rotations and homeomorphisms in a nontrivial Dehn-twist class, we prove uniqueness of the minimal set when the fibres are Jordan curves on a residual set of base parameters. The same conclusion holds if the fibre cores are Jordan curves, or if the fibres are locally connected, on a nonmeagre set of parameters. The residual hypothesis cannot be replaced by a full-measure hypothesis, even under area preservation and topological transitivity. For every totally irrational rotation vector, we construct such a map with Jordan-curve fibres almost everywhere and uncountably many pairwise disjoint uniquely ergodic minimal Cantor sets. We also construct examples in every nontrivial Dehn-twist class, with prescribed irrational vertical rotation number and bounded deviations. In both families, the Jordan-curve parameters form a meagre set of full Lebesgue measure.

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arXiv2026

Sharp Rainbow Path Covers in Dense and Complete Multipartite Graphs

A path in a properly edge-colored graph is rainbow if its edges have pairwise distinct colors. For a proper edge-coloring $c$ of a graph $G$, let $\operatorname{rpc}(G,c)$ be the minimum number of rainbow paths needed to cover $E(G)$, and let $\operatorname{rpc}(G)$ be the maximum of $\operatorname{rpc}(G,c)$ over all proper edge-colorings of $G$. We prove that, for every fixed $0<α<1$, every properly edge-colored $n$-vertex graph with minimum degree at least $αn$ satisfies $\operatorname{rpc}(G,c)\leq(1+o(1))n/2$, where the coefficient $1/2$ is best possible. We also determine $\operatorname{rpc}(G)$ asymptotically for every complete multipartite graph. If $G=K_{n_1,\ldots,n_r}$ has order $n$ and largest and smallest part sizes $M$ and $s$, respectively, then, uniformly over all choices of the number and sizes of the parts, $\operatorname{rpc}(G)=(1+o(1))\max\{\min\{\lfloor n/2\rfloor,n-M\},(n-s)/2\}$. The proof combines pseudorandom packings of globally rainbow linear forests with a decomposition into dense parts and prescribed avoidance for arbitrary dense graphs, and with reserved connectors and a direct dominant-part argument for complete multipartite graphs.

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arXiv2026

Strongly separating graph edges with $10n$ paths

A family of paths strongly separates the edges of a graph if every two distinct edges are separated in both directions by paths in the family. Bonamy, Botler, Dross, Naia, and Skokan proved that every $n$-vertex graph admits such a family of at most $19n$ paths. We improve this bound to $10n-o(n)$.

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arXiv2026

Sharp Same-Color Cycle Covers in Two-Colored Complete Graphs

We extend the conjecture of Erdős and Gyárfás on monochromatic path covers to the setting of monochromatic cycle covers. We prove that, for all $n$, every 2-edge-coloring of the complete graph on $n$ vertices contains a collection of at most $\lceil\sqrt{n}\rceil$ monochromatic cycles, all of the same color, that together cover all vertices. The order of the bound is best possible, and the ceiling is necessary for infinitely many $n$.

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arXiv2026

A rainbow version of Lehel's conjecture

Lehel's conjecture states that every 2-edge-colouring of K_n admits a partition of its vertex set into two monochromatic cycles. It was proven for sufficiently large n by Łuczak, Rödl, and Szemerédi in 1998, later improved by Allen in 2008, and fully resolved by Bessy and Thomassé in 2010. In this paper, we consider a rainbow analogue of Lehel's conjecture in the setting of properly edge-coloured complete graphs. We prove that, for sufficiently large n, every properly edge-coloured Kn admits a partition of its vertex set into two vertex-disjoint rainbow cycles

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arXiv2026

Linear Lower Bounds for the Modular Chromatic Index

Let $k\geq2$ be an integer. A $1\bmod k$ edge-coloring of a graph $G$ is an edge-coloring in which every nonzero degree in each color class is congruent to $1$ modulo $k$. Let $χ'_k(G)$ denote the minimum number of colors required, and let $χ'_k$ be the supremum of $χ'_k(G)$ over all finite simple graphs $G$. Botler, Colucci, and Kohayakawa conjectured that there exists an absolute constant $C$ such that $χ'_k(G)\leq k+C$ for every $k$ and every $G$. We disprove this conjecture, even within the class of bipartite graphs. More precisely, for all integers $c\geq0$ and $k\geq3c+2$, we construct a finite simple bipartite graph $G_{k,c}$ satisfying $χ'_k(G_{k,c})=k+c+1$. Consequently, $χ'_k\geq k+\lfloor(k+1)/3\rfloor$ for every $k\geq2$. For $k_m=2\cdot3^{m-1}$, we give an affine-hyperplane construction of a finite simple bipartite graph $G_m$ satisfying $Δ(G_m)=χ'_{k_m}(G_m)=3^m=3k_m/2$. More generally, for every sufficiently large $k$, we construct a finite simple bipartite graph $G_k$ such that $Δ(G_k)=χ'_k(G_k)\geq3k/2-10(k\log k)^{1/3}$. Our proofs combine a codegree obstruction with explicit cyclic and affine-geometric constructions and a structured random perturbation.

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arXiv2026

Large Monochromatic Components in Colored Random Graphs

We study the size of the largest monochromatic connected component that must appear in any edge-coloring of a random graph. Let $G\sim G(n,p)$ with $p\gg 1/n$ and $p=o(1)$, and write $np=he^h$. We show that, with high probability, every $2$-edge-coloring of $G$ contains a monochromatic connected component of order at least $n-Θ(ne^{-h})$. Moreover, we construct colorings showing that this bound is best possible up to constant factors. We extend this result to three colors: for $p\gg 1/n$ and $p=o(1)$, with high probability every $3$-edge-coloring of $G$ contains a monochromatic connected component of size at least $\frac{n}{2}-Θ(1/p)$, and this estimate is again tight up to constant factors. In the bipartite setting $G\sim G(n,n,p)$, under the same assumptions on $p$, we prove an analogous statement: with high probability, every $2$-edge-coloring contains two monochromatic components whose union covers all but $Θ(ne^{-h})$ vertices, and this bound is asymptotically sharp. Our approach is elementary and is based on analyzing large connected structures across suitably balanced vertex partitions.

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arXiv2026

Exact three-component covers in 2-coloured random bipartite graphs

We resolve the two-colour three-component conjecture of Fernández, Pavez-Signé and Stein for random bipartite graphs. More precisely, we prove that if $G\sim G(n,n,p)$ and $p\gg\sqrt{\log n/n}$, then with high probability every red--blue edge-colouring of $G$ admits a cover of its vertex set by at most three monochromatic connected components. The proof is based on a uniform expansion lemma for unions of common neighbourhoods and an alternating common-neighbourhood expansion argument.

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arXiv2026

On Turán Number of Graphs with Small Minimum Feedback Vertex Numbers

Given a graph $H$, the minimum feedback vertex number of $H$ is the minimum number of vertices whose removal results in an acyclic graph. In this paper, we investigate Turán-type extremal problems for bipartite graphs in terms of their feedback vertex number. Our first result concerns bipartite graphs $H$ with minimum feedback vertex number one. Such graphs can be obtained from a forest by identifying a specified collection of leaves into a single vertex. For these graphs, we show that $\text{ex}(n, H)$ is upper bounded by $O(n^{1+1/k^\ast})$, where $2k^\ast$ is the length of the shortest cycle contained in $H$. In addition, we consider a family of bipartite graphs with minimum feedback vertex number three. Let $E_{k,t}$ be the graph obtained from the theta graph $θ_{k,t}$ by joining a new vertex $x$ to one side of the bipartition and another vertex $y$ to the other. Let $E^+_{k,t}$ denote the graph obtained by adding the edge $xy$ to $E_{k,t}$. We prove that for any $k\geq 2$ and sufficiently large $t$, $\text{ex}(n, E^+_{k,t})= Θ(n^{\frac{3k-1}{2k-1}}).$

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arXiv2026

A Note on Weak Saturation Number of Trees

In this paper, we estimate the weak saturation numbers of trees. As a case study, we examine caterpillars and obtain several tight estimates. In particular, this implies that for any $α\in [1,2]$, there exist caterpillars with $k$ vertices whose weak saturation numbers are of order $k^α$. We call a tree good if its weak saturation number is exactly its edge number minus one. We provide a sufficient condition for a tree to be a good tree. With the additional property that all leaves are at even distances from each other, this condition fully characterizes good trees.

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arXiv2025

On the Color Discrepancy of Spanning Trees in Random and Randomly Perturbed Graphs

In this work, we study the color discrepancy of spanning trees in random graphs. We show that for the Erdős-Rényi random graph $G(n,p)$ with $p$ above the connectivity threshold, the following holds with high probability: in every 2-edge-coloring of the graph, there exists a spanning tree with a linear number of leaves such that one color class contains more than $\frac{1 + \varepsilon}{2}n $ of the tree's edges. Here, $\varepsilon>0$ is a small absolute constant independent of $p$. We also extend this line of research to randomly perturbed dense graphs, showing that adding a few random edges to a dense graph typically creates a spanning tree with a large color discrepancy under any 2-edge-coloring.

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arXiv2025

On Graph Odd Edge-Colorings and Odd Edge-Coverings

An odd $k$-edge-coloring of a graph $G$ is a (not necessarily proper) edge-coloring with at most $k$ colors such that each non-empty color class induces a graph in which every vertex is of odd degree; similarly, if more than one color per edge is allowed, we speak of an odd $k$-edge-covering of $G$. In this paper, we fully resolve two major conjectures on odd edge-colorings and odd edge-coverings of graphs, proposed by Petru{š}evski and {Š}krekovski ({\it European Journal of Combinatorics,} 91:103225, 2021). The first conjecture states that, apart from two particular exceptions which are respectively odd $5$- and odd-$6$-edge-colorable, for any other loopless and connected graph $G$ there exists an edge $e$ such that $G\backslash \{e\}$ is odd $3$-edge-colorable. The second conjecture states that any simple graph $G$ admits an odd $3$-edge-covering in which at most one edge receives more than one color. In addition, we strongly confirm the second conjecture by demonstrating that there exists an odd $3$-edge-covering in which at most one edge receives two colors and the rest of the edges receive unique colors.

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arXiv2025

Turán Number of Subdivisions of Multipartite Graphs

In this paper, we investigate the Turán exponent for $1$-subdivisions of graphs that are neither bipartite nor complete. Specifically, we establish an upper bound on the Turán number of the 1-subdivision of $K_{s,t}^+$, where $K_{s,t}^+$ is obtained by adding a single edge within the part of size $s$ of the complete bipartite graph $K_{s,t}$, with $4\leq s \leq t$. In addition, we derive an upper bound for the extremal number of a family of graphs formed by (possibly degenerate) 1-subdivisions of certain tripartite graphs.

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arXiv2023

More on Rainbow Cliques in Edge-Colored Graphs

In an edge-colored graph $G$, a rainbow clique $K_k$ is a $k$-complete subgraph in which all the edges have distinct colors. Let $e(G)$ and $c(G)$ be the number of edges and colors in $G$, respectively. In this paper, we show that for any $\varepsilon>0$, if $e(G)+c(G) \geq (1+\frac{k-3}{k-2}+2\varepsilon) {n\choose 2}$ and $k\geq 3$, then for sufficiently large $n$, the number of rainbow cliques $K_k$ in $G$ is $Ω(n^k)$. We also characterize the extremal graphs $G$ without a rainbow clique $K_k$, for $k=4,5$, when $e(G)+c(G)$ is maximum. Our results not only address existing questions but also complete the findings of Ehard and Mohr (Ehard and Mohr, Rainbow triangles and cliques in edge-colored graphs. {\it European Journal of Combinatorics, 84:103037,2020}).

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arXiv2023

On the Turán Number of Generalized Theta Graphs

Let $Θ_{k_1,\cdots,k_\ell}$ denote the generalized theta graph, which consists of $\ell$ internally disjoint paths with lengths $k_1,\cdots, k_{\ell}$, connecting two fixed vertices. We estimate the corresponding extremal number $\text{ex}(n,Θ_{k_1,\cdots,k_\ell})$. When the lengths of all paths have the same parity and at most one path has length 1, $\text{ex}(n,Θ_{k_1,\cdots,k_\ell})$ is $O(n^{1+1/k^\ast})$, where $2k^\ast$ is the length of the smallest cycle in $Θ_{k_1,\cdots,k_\ell}$. We also establish matching lower bound in the particular case of $\text{ex}(n,Θ_{3,5,5})$.

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arXiv2022

On Non-contractible Periodic Orbits and Bounded Deviations

We present a dichotomy for surface homeomorphisms in the isotopy class of the identity. We show that, in the absence of a degenerate fixed point set, either there exists a uniform bound on the diameter of orbits of non-wandering points for the lifted dynamics in the universal covering space, or the map has non-contractible periodic points. We then use this new tool to characterize the dynamics of area preserving homeomorphisms of the torus without non-contractible periodic points, showing that if the fixed point set is non-degenerate, then either the lifted dynamics is uniformly bounded, or the lifted map has a single strong irrational dynamical direction.

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arXiv2022

On Color Isomorphic Pairs in Proper Edge Colourings of Complete Graphs

Following the recent paper which initiated the study of colour isomorphism problems for complete graphs, we obtain upper bounds for $f_2(n,H)$ for a family of graphs $H$ obtained as the $K_0$-th rooted power of a balanced rooted tree for some sufficiently large $K_0$. The proof uses the random polynomial method of Bukh. We also obtain matching lower bounds for $1$-subdivisions of the complete bipartite graph.

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arXiv2021

On Stable and Unstable Behaviours of Certain Rotation Segments

In this paper, we study non-wandering homeomorphisms of the two torus in the identity homotopy class, whose rotation sets are non-trivial line segments from $(0,0)$ to some totally irrational vector $(α,β)$. We show this rotation set is in fact a non-generic phenomenon for any $C^r$ diffeomorphisms, with $r \geq 1$. When such a rotation set does happen, assuming several natural conditions that are generically satisfied in the area-preserving world, we give a clearer description of its rotational behavior. More precisely, the dynamics admits bounded deviation along the direction $-(α,β)$ in the lift, and the rotation set is locked inside an arbitrarily small cone with respect to small $C^0$-perturbations of the dynamics. On the other hand, for any non-wandering homeomorphism $f$ with this kind of rotation set, we also present a perturbation scheme in order for the rotation set to be eaten by rotation sets of nearby dynamics, in the sense that the later set has non-empty interior and contains the former one. These two flavors interplay and share the common goal of understanding the stability/instability properties of this kind of rotation set.

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