arXiv subjects

Xiamiao Zhao

Publications and source records attributed to Xiamiao Zhao.

31 records · Page 2Linked to original sources

arXiv2026

Forbidding edge-critical graphs as trace in uniform hypergraphs

We say a hypergraph $\mathcal{H}$ contains a graph $G$ as trace if there exists a vertex subset $S \subseteq V(\mathcal{H})$ such that $|S| = V(G)$ and $\{e \cap S \mid e \in E(\mathcal{H})\}$ contains $G$ as a subgraph. We use $\mathrm{ex}(n, Tr_r(G))$ to denote the maximum number of edges in an $r$-uniform hypergraph on $n$ vertices not containing $G$ as trace. The study of Turán numbers for traces was initiated by Mubayi and Zhao~(2017) who studied $\mathrm{ex}(n, Tr_r(K_{s+1}))$ where $K_{s+1}$ is a clique on $s+1$ vertices and conjectured the exact value of $\mathrm{ex}(n, Tr_r(K_{s+1}))$. When $r \le s$, the conjecture was covered by a result of Pikhurko~(2013) who gave the exact value of Turán numbers for expanded cliques. Then Gerbner and Picollelli~(2023) gave the exact value for book graphs~($K_{1,1,t}$, the complete tripartite graph with two parts of size one and one part of size $t \ge 2$). We say $G$ is edge-critical if there exists an edge $e \in E(G)$ such that $χ(G - e) < χ(G)$ where $χ(G)$ is the chromatic number of $G$. The definition of edge-critical was given by Simonovits~(1974), who proved that for an edge-critical graph $G$ with $χ(G) = s+1 \ge 3$, the Turán graph $T(n,s)$ is the unique extremal graph for $ex(n,G)$ as $n$ is sufficiently large. In this paper, we further generalize the results of Gerbner and Picollelli~(2023) to edge-critical graphs. More precisely, we prove that for an edge-critical graph $G$ with $χ(G) = s+1$, when $s \ge r \ge 3$ and $n$ is sufficiently large, the $r$-uniform Turán graph $T_r(n,s)$ is the unique extremal hypergraph.

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arXiv2026

The Turán number of Berge matchings

Given a graph $F$, an $r$-uniform hypergraph $\mathcal{H}$ is a {\em Berge-$F$} if there is a bijection $ϕ:E(F)\to E(\mathcal{H})$ such that $e\subseteq ϕ(e)$ for each $e\in E(F)$. Given a family $\mathcal{F}$ of $r$-uniform hypergraphs, an $r$-uniform hypergraph is $\mathcal{F}$-free if it does not contain any member of $\mathcal{F}$ as a subhypergraph. The Turán number of $\mathcal{F}$ is the maximum number of hyperedges in an $\mathcal{F}$-free $r$-graph on $n$ vertices. Let $M_{s+1}$ denote a matching of size $s+1$, i.e., the graph consisting of $s+1$ independent edges. Khormali and Palmer [\textit{European J. Combin.} 102 (2022) 103506] completely determined the Turán number of Berge matchings for sufficiently large $n$. Subsequently, Kang, Ni, and Shan [\textit{Discrete Math.} 345 (2022) 112901] determined the exact value of the Turán number of Berge-$M_{s+1}$ for all $n$ when $r \le s-1$ or $r \ge 2s+2$. In this paper, we settle the final open case $s \le r \le 2s+1$, thereby completing the determination of the Turán number of Berge matchings.

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arXiv2026

Extremal results on Berge disjoint paths

The well-known Erdős-Gallai Theorem gave the Turán number of paths. Bushaw and Kettle generalized this result to consider the Turán number of disjoint paths. Since then, many studies are focused on the Turán number of linear forest. For a graph $F$, an $r$-uniform hypergraph $\mathcal{H}$ is a $\text{Berge-} F$ if there is a bijection $ϕ: E(F)\to E(\mathcal{H})$ such that $e\subseteq ϕ(e)$ for each $e\in E(F)$. When $F$ is a path, we call $\text{Berge-} F$ a Berge path. The Turán number of Berge paths was initially studied by Győri, Katona and Lemons. They gave the value of $\text{ex}_r(n,\text{Berge-}P_\ell)$ for $\ell>r+1$. This result is a generalization of Erdős-Galli Theorem. Since then, the Turán number of Berge paths has received widespread attention. Recently, Zhou, Gerbner and Yuan initially studied the Turán number of Berge disjoint paths and for the cases when all the paths have odd length. In this paper, we give a more general result, which gives the exact value of $\mathrm{ex}_r(n,\text{Berge-} kP_{\ell})$ for all $k\geq 2$, $r\ge 3$, and $\ell\geq r+7$.

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arXiv2026

On generalized Turán problems for expansions

Given a graph $F$, the $r$-expansion $F^r$ of $F$ is the $r$-uniform hypergraph obtained from $F$ by inserting $r-2$ new distinct vertices in each edge of $F$. Given $r$-uniform hypergraphs $\mathcal{H}$ and $\mathcal{F}$, the generalized Turán number, denoted by $\textrm{ex}_r(n,\mathcal{H},\mathcal{F})$, is the maximum number of copies of $\mathcal{H}$ in an $n$-vertex $r$-uniform hypergraph that does not contain $\mathcal{F}$ as a subhypergraph. In the case where $r=2$ (i.e., the graph case), the study of generalized Turán problems was initiated by Alon and Shikhelman [\textit{J. Combin. Theory Series B.} 121 (2016) 146--172]. Motivated by their work, we systematically study generalized Turán problems for expansions and obtain several general and exact results. In particular, for the non-degenerate case, we determine the exact generalized Turán number for expansions of complete graphs, and establish the asymptotics of the generalized Turán number for expansions of the vertex-disjoint union of complete graphs. For the degenerate case, we establish the asymptotics of generalized Turán numbers for expansions of several classes of forests, including star forests, linear forests and star-path forests.

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arXiv2025

The minimum edge-pancyclic graph of a given order

A graph $G$ of order $n$ is called edge-pancyclic if, for every integer $k$ with $3 \leq k \leq n$, every edge of $G$ lies in a cycle of length $k$. Determining the minimum size $f(n)$ of a simple edge-pancyclic graph with $n$ vertices seems difficult. Recently, Li, Liu and Zhan \cite{li2024minimum} gave both a lower bound and an upper bound of $f(n)$. In this paper, we improve their lower bound by considering a new class of graphs and improve the upper bound by constructing a family of edge-pancyclic graphs.

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arXiv2025

Edge version of the inducibility via the entropy method

The inducibility of a graph $H$ is about the maximum number of induced copies of $H$ in a graph on $n$ vertices. We consider its edge version, that is, the maximum number of induced copies of $H$ in a graph with $m$ edges. Let $c(G,H)$ be the number of induced copies of $H$ in $G$ and $ρ(H,m) = \max \{c(G,H) \mid |E(G)| = m\}$. For any graph $H$, we prove that $ρ(H,m) = Θ(m^{α_f(H)})$ where $α_f(H)$ is the fractional independence number of $H$. Therefore, we now focus on the constant factor in front of $m^{α_f(H)}$. In this paper, we give some results of $ρ(H,m)$ when $H$ is a cycle or path. We conjecture that for any cycle $C_k$ with $k \ge 5$, $ρ(C_k,m)= (1+o(1))\left( m/k\right)^{k/2}$ and the bound achieves by the blow up of $C_k$. For even cycles, we establish an upper bound with an extra constant factor. For odd cycles, we can only establish an upper bound with an extra factor depending on $k$. We prove that $ρ(P_{2l},m) \le \frac{m^l}{2(l-1)^{l-1}}$ and $ρ(P_{2l+1},m) \le \frac{m^{l+1}}{4l^l}$, where $l \ge 2$. We also conjecture the asymptotic value of $ρ(P_k, m)$. The entropy method is mainly used to prove our results.

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arXiv2025

The maximum sum of sizes of non-empty cross $L$-intersecting families

Let $n$, $r$, and $k$ be positive integers such that $k, r \geq 2$, $L$ a non-empty subset of $[k]$, and $\mathcal{F}_i \subseteq \binom{[n]}{k}$ for $1 \leq i \leq r$. We say that non-empty families $\mathcal{F}_1, \mathcal{F}_2, \ldots, \mathcal{F}_r$ are $r$-cross $L$-intersecting if $\left| \bigcap_{i=1}^r F_i \right| \in L$ for every choice of $F_i \in \mathcal{F}_i$ with $1 \leq i \leq r$. They are called pairwise cross $L$-intersecting if $|A \cap B| \in L$ for all $A \in \mathcal{F}_i$, $B \in \mathcal{F}_j$ with $i \neq j$. If $r=2$, we simply say cross $L$-intersecting instead of $2$-cross $L$-intersecting or pairwise cross $L$-intersecting. In this paper, we determine the maximum possible sum of sizes of non-empty cross $L$-intersecting families $\mathcal{F}_1$ and $\mathcal{F}_2$ for all admissible $n$, $k$, and $L$, and we characterize all the extremal structures. We also establish the maximum value of the sum of sizes of families $\mathcal{F}_1, \dots, \mathcal{F}_r$ that are both pairwise cross $L$-intersecting and $r$-cross $L$-intersecting, provided $n$ is sufficiently large and $L$ satisfies certain conditions. Furthermore, we characterize all such families attaining the maximum total size.

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arXiv2025

Counting induced subgraphs with given intersection sizes

Let $F$ be a graph of order $r$. In this paper, we study the maximum number of induced copies of $F$ with restricted intersections, which highlights the motivation from extremal set theory. Let $L=\{\ell_1,\dots,\ell_s\}\subseteq[0,r-1]$ be an integer set with $s\not\in\{1,r\}$. Let $Ψ_r(n,F,L)$ be the maximum number of induced copies of $F$ in an $n$-vertex graph, where the induced copies of $F$ are $L$-intersecting as a family of $r$-subsets, i.e., for any two induced copies of $F$, the size of their intersection is in $L$. Helliar and Liu initiated a study of the function $Ψ_r(n,K_r,L)$. Very recently, Zhao and Zhang improved their result and showed that $Ψ_r(n,K_r,L)=Θ_{r,L}(n^{s})$ if and only if $\ell_1,\dots,\ell_s,r$ form an arithmetic progression. In this paper, we show that $Ψ_r(n,F,L)=o_{r,L}(n^{s})$ when $\ell_1,\dots,\ell_s,r$ do not form an arithmetic progression. We study the asymptotical result of $Ψ_r(n,C_r,L)$, and determined the asymptotically optimal result when $\ell_1,\dots,\ell_s,r$ form an arithmetic progression and take certain values. We also study the generalized Turán problem, determining the maximum number of $H$, where the copies of $H$ are $L$-intersecting as a family of $r$-subsets. The entropy method is used to prove our results.

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arXiv2025

Edge pancyclic Cayley graphs on symmetric group

We study the derangement graph $Γ_n$ whose vertex set consists of all permutations of $\{1,\ldots,n\}$, where two vertices are adjacent if and only if their corresponding permutations differ at every position. It is well-known that $Γ_n$ is a Cayley graph, Hamiltonian and Hamilton-connected. In this paper, we prove that for $n \geq 4$, the derangement graph $Γ_n$ is edge pancyclic. Moreover, we extend this result to two broader classes of Cayley graphs defined on symmetric group.

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arXiv2025

Turán type problems for a fixed graph and a linear forest

Let $\mathscr{F}$ be a family of graphs. A graph $G$ is $\mathscr{F}$-free if $G$ does not contain any $F\in \mathscr{F}$ as a subgraph. The Turán number, denoted by $ex(n, \mathscr{F})$, is the maximum number of edges in an $n$-vertex $\mathscr{F}$-free graph. Let $F $ be a fixed graph with $ χ(F) \geq 3 $. A forest $H$ is called a linear forest if all components of $H$ are paths. In this paper, we determined the exact value of $ex(n, \{H, F\}) $ for a fixed graph $F$ with $χ(F)\geq 3$ and a linear forest $H$ with at least $2$ components and each component with size at least $3$.

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arXiv2024

Generalized Turán problems for a matching and long cycles

Let $\mathscr{F}$ be a family of graphs. A graph $G$ is $\mathscr{F}$-free if $G$ does not contain any $F\in \mathcal{F}$ as a subgraph. The general Turán number, denoted by $ex(n, H,\mathscr{F})$, is the maximum number of copies of $H$ in an $n$-vertex $\mathscr{F}$-free graph. Then $ex(n, K_2,\mathscr{F})$, also denote by $ex(n, \mathscr{F})$, is the Turán number. Recently, Alon and Frankl determined the exact value of $ex(n, \{K_{k},M_{s+1}\})$, where $K_{k}$ and $M_{s+1}$ are a complete graph on $k $ vertices and a matching of size $s +1$, respectively. Then many results were obtained by extending $K_{k}$ to a general fixed graph or family of graphs. Let $C_k$ be a cycle of order $k$. Denote $C_{\ge k}=\{C_k,C_{k+1},\ldots\}$. In this paper, we determine the value of $ex(n,K_r, \{C_{\ge k},M_{s+1}\})$ for large enough $n$ and obtain the extremal graphs when $k$ is odd. Particularly, the exact value of $ex(n, \{C_{\ge k},M_{s+1}\})$ and the extremal graph are given for large enough $n$.

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arXiv2024

The Minimum Weighting Ratio Problem and Its Application in Chordal Graphs

Constructing the maximum spanning tree $T$ of an edge-weighted connected graph $G$ is one of the important research topics in computer science and optimization, and the related research results have played an active role in practical applications. In this paper, we are concerned with the ratio of the weighted sum of a spanning tree $T$ of $G$ to the weighted sum of $G$, which we try to minimize. We propose an interesting theorem to simplify this problem and show that this optimal problem can be solved in polynomial time. Furthermore, we apply the optimal problem in chordal graphs.

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arXiv2024

Turán number of complete bipartite graphs with bounded matching number

Let $\mathscr{F}$ be a family of graphs. A graph $G$ is $\mathscr{F}$-free if $G$ does not contain any $F\in \mathcal{F}$ as a subgraph. The Turán number $ex(n, \mathscr{F})$ is the maximum number of edges in an $n$-vertex $\mathscr{F}$-free graph. Let $M_{s}$ be the matching consisting of $ s $ independent edges. Recently, Alon and Frank determined the exact value of $ex(n,\{K_{m},M_{s+1}\})$. Gerbner obtained several results about $ex(n,\{F,M_{s+1}\})$ when $F$ satisfies certain proportions. In this paper, we determine the exact value of $ex(n,\{K_{l,t},M_{s+1}\})$ when $s, n$ are large enough for every $3\leq l\leq t$. When $n$ is large enough, we also show that $ex(n,\{K_{2,2}, M_{s+1}\})=n+{s\choose 2}-\left\lceil\frac{s}{2}\right\rceil$ for $s\ge 12$ and $ex(n,\{K_{2,t},M_{s+1}\})=n+(t-1){s\choose 2}-\left\lceil\frac{s}{2}\right\rceil$ when $t\ge 3$ and $s$ is large enough.

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