arXiv subjects
Publications and source records attributed to Weichan Liu.
We prove that every $n$-vertex simple planar graph with no copy of $C_8$ has at most \[ \frac{69}{25}(n-2) \] edges, for every $n\ge 8$. This improves the best known bound \[ \frac{323}{108}n-6 \qquad \text{for every } n\ge 27. \]
For $k$-graphs $F$ and $H_0$ the $F$-bootstrap percolation process (or $F$-process) starting with $H_0$ is a sequence $(H_i)_{i\geq0}$ of $k$-graphs such that $H_{i+1}$ is obtained from $H_i$ by adding all those $e\in V(H_0)^{(k)}\setminus E(H_i)$ as edges that complete a new copy of $F$. The running time of this $F$-process, denoted by $M_F(H_0)$, is the smallest $i$ with $H_i=H_{i+1}$. Bollob\'as proposed the problem of determining the maximum running time for $n\in\mathbb{N}$, i.e., $M_F(n)=\max_{\vert V(H_0)\vert=n}M_F(H_0)$. Although this problem has received a lot of attention recently, until now the best known upper bound for $M_{K_t}(n)$, with $t\geq5$, was the trivial bound $\binom{n}{2}$. Here we provide the first non-trivial upper bound for this problem by showing that $$M_{K_t}(n)\leq\Big(\frac{t-3}{t-2}+o(1)\Big)\binom{n}{2}$$ holds for every integer $t\geq 3$. In fact, we prove the following more general result. For every $k\geq2$, every $k$-graph $F$, and every $e\in E(F)$ we have $M_F(n)\leq\big(\pi(F-e)+o(1)\big)\binom{n}{k}$, where $\pi$ is the Tur\'an density.
For $k$-graphs $F$ and $H_0$ the $F$-bootstrap percolation process (or $F$-process) starting with $H_0$ is a sequence $(H_i)_{i\geq0}$ of $k$-graphs such that $H_{i+1}$ is obtained from $H_i$ by adding all those $e\in V(H_0)^{(k)}\setminus E(H_i)$ as edges that complete a new copy of $F$. The running time of this $F$-process, denoted by $M_F(H_0)$, is the smallest $i$ with $H_i=H_{i+1}$. Bollob\'as proposed the problem of determining the maximum running time for $n\in\mathbb{N}$, i.e., $$M_F(n)=\max_{\vert V(H_0)\vert=n}M_F(H_0)\,.$$ Recently, Noel and Ranganathan initiated the study of this quantity for $k$-graphs. In this work, we determine the asymptotics of $M_F(n)$ for a large class of $k$-graphs. Given a graph $G=(V,E)$, the $k$-extension of $G$ is a $k$-graph $F^{(k)}(G)$ obtained from $G$ by enlarging each edge with a $(k-2)$-set of new vertices. We show that for every graph $G$ on $t$ vertices and every $k\geq 3$, $M_{F^{(k)}(G)}(n)\leq C_{k,t}$ for some constant $C_{k,t}$ depending only on $t$ and $k$.
A connected subgraph of a graph is isometric if it preserves distances. In this short note, we provide counterexamples to several variants of the following general question: When a graph $G$ is edge covered by connected isometric subgraphs $H_1,\dots,H_k$, which properties of $G$ can we infer from properties of $H_1,\dots,H_k$? For example, Dumas, Foucaud, Perez and Todinca (SIDMA, 2024) proved that when $H_1,\dots,H_k$ are paths, then the pathwidth of $G$ is bounded in terms of $k$. Among others, we show that there are graphs of arbitrarily large treewidth that can be isometrically edge covered by four trees.
Denote by $T_k$ the generalised triangle, a $k$-uniform hypergraph on vertex set $\{1,2,\dots,2k-1\}$ with three edges $\{1,\dots,k-1,k\}$,$\{1,\dots,k-1,k+1\}$ and $\{k,k+1,\dots,2k-1\}$. Recently, Bowtell, Kathapurkar, Morrison and Mycroft [arXiv: 2505.05606] established the exact minimum codegree threshold for perfect $T_3$-tilings in $3$-graphs. In this paper, we extend their result to all $k \geq 3$, determining the optimal minimum codegree threshold for perfect $T_k$-tilings in $k$-graphs. Our proof uses the lattice-based absorption method, as is usual, but develops a unified and effective approach to build transferrals for all uniformities, which is of independent interest. Additionally, we establish an asymptotically tight minimum codegree threshold for a rainbow variant of the problem.
In this paper, we generalize the concepts related to rainbow coloring to hypergraphs. Specifically, an $(n,r,H)$-local coloring is defined as a collection of $n$ edge-colorings, $f_v: E(K^{(r)}_n) \rightarrow [k]$ for each vertex $v$ in the complete $r$-uniform hypergraph $K^{(r)}_n$, with the property that for any copy $T$ of $H$ in $K^{(r)}_n$, there exists at least one vertex $u$ in $T$ such that $f_u$ provides a rainbow edge-coloring of $T$ (i.e., no two edges in $T$ share the same color under $f_u$). The minimum number of colors required for this coloring is denoted as the local rainbow coloring number $C_r(n, H)$. We first establish an upper bound of the local rainbow coloring number for $r$-uniform hypergraphs $H$ consisting of $h$ vertices, that is, $C_r(n, H)= O\left( n^{\frac{h-r}{h}} \cdot h^{2r + \frac{r}{h}} \right)$. Furthermore, we identify a set of $r$-uniform hypergraphs whose local rainbow coloring numbers are bounded by a constant. A notable special case indicates that $C_3(n,H) \leq C(H)$ for some constant $C(H)$ depending only on $H$ if and only if $H$ contains at most 3 edges and does not belong to a specific set of three well-structured hypergraphs, possibly augmented with isolated vertices. We further establish two 3-uniform hypergraphs $H$ of particular interest for which $C_3(n,H) = n^{o(1)}$. Regarding lower bounds, we demonstrate that for every $r$-uniform hypergraph $H$ with sufficiently many edges, there exists a constant $b = b(H) > 0$ such that $C_r(n,H) = \Omega(n^b)$. Additionally, we obtain lower bounds for several hypergraphs of specific interest.
An equitable coloring of a graph is a proper coloring where the sizes of any two different color classes do not differ by more than one. A graph is IC-planar if it can be drawn in the plane so that no two crossed edges have a common endpoint, and is NIC-planar graphs if it can be embedded in the plane in such a way that no two pairs of crossed edges share two endpoints. Zhang proved that every IC-planar graph with maximum degree $\Delta\geq 12$ and every NIC-planar graph with maximum degree $\Delta\geq 13$ have equitable $\Delta$-colorings. In this paper, we reduce the threshold from 12 to 10 for IC-planar graphs and from 13 to 11 for NIC-planar graphs.
In an $r$-partite graph, an independent transversal of size $s$ (ITS) consists of $s$ vertices from each part forming an independent set. Motivated by a question from Bollob\'as, Erd\H{o}s, and Szemer\'edi (1975), Di Braccio and Illingworth (2024) inquired about the minimum degree needed to ensure an $n \times \cdots \times n$ $r$-partite graph contains $K_r(s)$, a complete $r$-partite graph with $s$ vertices in each part. We reformulate this as finding the smallest $n$ such that any $n \times \cdots \times n$ $r$-partite graph with maximum degree $\Delta$ has an ITS. For any $\varepsilon>0$, we prove the existence of a $\gamma>0$ ensuring that if $G$ is a multipartite graph partitioned as $(V_1, V_2, \ldots, V_r)$, where the average degree of each part $V_i$ is at most $D$, the maximum degree of any vertex to any part $V_i$ is at most $\gamma D$, and the size of each part $V_i$ is at least $(s + \varepsilon)D$, then $G$ possesses an ITS. The constraint $(s + \varepsilon)D$ on the part size is tight. This extends results of Loh and Sudakov (2007), Glock and Sudakov (2022), and Kang and Kelly (2022). We also show that any $n \times \cdots \times n$ $r$-partite graph with minimum degree at least $\left(r-1-\frac{1}{2s^2}\right)n$ contains $K_r(s)$ and provide a relative Tur\'an-type result. Additionally, this paper explores counting ITSs in multipartite graphs.
The codegree Tur\'an density $\gamma(F)$ of a $k$-graph $F$ is the smallest $\gamma\in[0,1)$ such that every $k$-graph $H$ with $\delta_{k-1}(H)\geq(\gamma+o(1))\vert V(H)\vert$ contains a copy of $F$. We prove that for all $k,r\in\mathbb{N}$ with $k\geq3$, $\frac{r-1}{r}$ is an accumulation point of $\Gamma^{(k)}=\{\gamma(F):F\text{ is a }k\text{-graph}\}$. This makes progress on a problem posed by Mubayi and Zhao.
An equitable coloring of a graph is a proper coloring where the sizes of any two distinct color classes differ by at most one. The celebrated Chen-Lih-Wu Conjecture (CLWC for short) states that every connected graph $G$ that is neither an odd cycle, a $K_r$, nor a $K_{2m+1,2m+1}$ has an equitable $\Delta(G)$-coloring. A graph $G$ is in $\mathcal{G}_{m_1,m_2}$ if for all $H\subseteq G$, $\lVert H \rVert\leq m_1|H|$, and if $H$ is bipartite, then $\lVert H \rVert\leq m_2|H|$. In this paper, we confirm CLWC for all graphs $G$ in $\mathcal{G}_{m_1, m_2}$ provided that $m_1\leq 1.8m_2$ and $\Delta(G)\geq \frac{2m_1}{1-\beta}$, where $\beta$ is a real root of $2m_2(1-x)(1+x)^2-m_1x(2+x)$. By specializing to the case $m_1 = m_2 = d$, we deduce that every $d$-degenerate graph $G$ with $\Delta(G) \geq 6.21d$ admits an equitable $r$-coloring for all $r \geq \Delta(G)$, thereby improving the previous best-known lower bound of $10d$ on $\Delta(G)$ established by Kostochka and Nakprasit in 2005. A graph is $k$-planar if it can be drawn in the plane so that each edge is crossed at most $k$ times. CLWC had been confirmed for planar graphs $G$ with $\Delta(G) \geq 8$ (Kostochka, Lin, and Xiang, 2024) and for $1$-planar graphs $G$ with $\Delta(G) \geq 13$ (Cranston and Mahmoud, 2025). As an immediate application of our main result, we extend this confirmation to all $k$-planar graphs $G$ with $k \geq 2$ and $\Delta(G) \geq \sqrt{383k}$.
Given a class $\mathcal{H}$ of $m$ hypergraphs ${H}_1, {H}_2, \ldots, {H}_m$ with the same vertex set $V$, a cooperative coloring of them is a partition $\{I_1, I_2, \ldots, I_m\}$ of $V$ in such a way that each $I_i$ is an independent set in ${H}_i$ for $1\leq i\leq m$. The cooperative chromatic number of a class $\mathcal{H}$ is the smallest number of hypergraphs from $\mathcal{H}$ that always possess a cooperative coloring. For the classes of $k$-uniform tight cycles, $k$-uniform loose cycles, $k$-uniform tight paths, and $k$-uniform loose paths, we find that their cooperative chromatic numbers are all exactly two utilizing a new proved set system partition theorem, which also has its independent interests and offers a broader perspective. For the class of $k$-partite $k$-uniform hypergraphs with sufficient large maximum degree $d$, we prove that its cooperative chromatic number has lower bound $\Omega(\log_k d)$ and upper bound $\text{O}\left(\frac{d}{\ln d}\right)^{\frac{1}{k-1}}$.
In this paper, we introduce the notion of 2-boundary planar graphs. A graph is 2-boundary planar if it has an embedding in the plane so that all vertices lie on the boundary of at most two faces and no edges are crossed. A proper coloring of a graph is odd if every non-isolated vertex has some color that appears an odd number of times on its neighborhood. Petru\v{s}evski and \v{S}krekovski conjectured in 2022 that every planar graph admits an odd 5-coloring. We confirm this conjecture for 2-boundary planar graphs. Moreover, we present several questions regarding 2-boundary planar graphs that are of independent interest.
An incidence of a hypergraph $\mathcal{H}=(X,S)$ is a pair $(x,s)$ with $x\in X$, $s\in S$ and $x\in s$. Two incidences $(x,s)$ and $(x',s')$ are adjacent if (i) $x=x'$, or (ii) $\{x,x'\}\subseteq s$ or $\{x,x'\}\subseteq s'$. A proper incidence $k$-coloring of a hypergraph $\mathcal{H}$ is a mapping $φ$ from the set of incidences of $\mathcal{H}$ to $\{1,2,\ldots,k\}$ so that $φ(x,s)\neq φ(x',s')$ for any two adjacent incidences $(x,s)$ and $(x',s')$ of $\mathcal{H}$. The incidence chromatic number $χ_I(\mathcal{H})$ of $\mathcal{H}$ is the minimum integer $k$ such that $\mathcal{H}$ has a proper incidence $k$-coloring. In this paper we prove $χ_I(\mathcal{H})\leq (4/3+o(1))r(\mathcal{H})Δ(\mathcal{H})$ for every $t$-quasi-linear hypergraph with $t<<r(\mathcal{H})$ and sufficiently large $Δ(\mathcal{H})$, where $r(\mathcal{H})$ is the maximum of the cardinalities of the edges in $\mathcal{H}$. It is also proved that $χ_I(\mathcal{H})\leq Δ(\mathcal{H})+r(\mathcal{H})-1$ if $\mathcal{H}$ is an $α$-acyclic linear hypergraph, and this bound is sharp.
A signed tree-coloring of a signed graph $(G,σ)$ is a vertex coloring $c$ so that $G^{c}(i,\pm)$ is a forest for every $i\in c(u)$ and $u\in V(G)$, where $G^{c}(i,\pm)$ is the subgraph of $(G,σ)$ whose vertex set is the set of vertices colored by $i$ or $-i$ and edge set is the set of positive edges with two end-vertices colored both by $i$ or both by $-i$, along with the set of negative edges with one end-vertex colored by $i$ and the other colored by $-i$. If $c$ is a function from $V(G)$ to $M_n$, where $M_n$ is $\{\pm 1,\pm 2,\ldots,\pm k\}$ if $n=2k$, and $\{0,\pm 1,\pm 2,\ldots,\pm k\}$ if $n=2k+1$, then $c$ a signed tree-$n$-coloring of $(G,σ)$. The minimum integer $n$ such that $(G,σ)$ admits a signed tree-$n$-coloring is the signed vertex arboricity of $(G,σ)$, denoted by $va(G,σ)$. In this paper, we first show that two switching equivalent signed graphs have the same signed vertex arboricity, and then prove that $va(G,σ)\leq 3$ for every balanced signed triangulation and for every edge-maximal $K_5$-minor-free graph with balanced signature. This generalizes the well-known result that the vertex arboricity of every planar graph is at most 3.