arXiv subjects

Xiamiao Zhao

Publications and source records attributed to Xiamiao Zhao.

At least 19 recordsLinked to original sources

arXiv2026

Tur\'an-good monotonicity thresholds

A graph $H$ is $K_{r+1}$-Tur\'an-good if, for every sufficiently large $n$, the Tur\'an graph $T_r(n)$ maximizes the number of copies of $H$ among all $n$-vertex $K_{r+1}$-free graphs. It is strictly $K_{r+1}$-Tur\'an-good if $T_r(n)$ is the unique extremal graph. Morrison, Nir, Norin, Rz\k{a}\.zewski and Wesolek [\emph{JCTB}, 2023] proved that every graph $H$ is $K_{r+1}$-Tur\'an-good whenever $r\ge 300v(H)^9$. They raised the following two questions: 1.Can the sufficient condition $r\ge 300v(H)^9$ be reduced to a condition of quadratic order in $v(H)$? 2.Is the Tur\'an-good property monotone in $r$? More precisely, if a graph $H$ is $K_r$-Tur\'an-good, must it also be $K_{r+1}$-Tur\'an-good? We affirmatively resolve the first question and derive an even stronger bound linear in the edge number: every graph $H$ with at least one edge is strictly $K_{r+1}$-Tur\'an-good and $K_{r+1}$-Tur\'an-stable whenever $r\ge 168e(H)$. This condition is quadratic in $v(H)$ for arbitrary graphs and linear in $v(H)$ for every sparse graph family with $e(H)=O(v(H))$. We answer the second question negatively. For every $r\ge3$, there exists a graph that is strictly $K_r$-Tur\'an-good but not $K_{r+1}$-Tur\'an-good. More quantitatively, for every sufficiently large $h$, there exists a graph $H$ with $v(H)\le h$ and an integer $r=h-2\sqrt h+O(1)$ such that $H$ is strictly $K_r$-Tur\'an-good but not $K_{r+1}$-Tur\'an-good. The monotonicity threshold $\lambda(H)$ is the least integer $R\ge 2$ such that, for every $r\ge R$, the graph $H$ is $K_{r+1}$-Tur\'an-good whenever it is $K_r$-Tur\'an-good. For \[ \lambda_{\max}(h)=\max\{\lambda(H)\mid v(H)\le h\}, \] our two results yield \[ h-2\sqrt h-O(1)\le \lambda_{\max}(h)\le 84h^2. \]

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arXiv2026

A stability theorem for Berge Hamiltonian cycles under a minimum degree condition

In this paper, we study extremal and stability problems for Berge Hamiltonian cycles in $r$-uniform hypergraphs under a minimum degree condition. Let $ g_r(n,t)=\binom{n-t}{r}+t\binom{t}{r-1}$, and let $t=t(k)$ be the unique integer satisfying $\binom{t-1}{r-1}<k\le \binom{t}{r-1}$. Using a sharp P\'osa-type degree sequence theorem of Salia, we prove an extremal upper bound on the number of hyperedges in an $n$-vertex $r$-uniform hypergraph with minimum degree at least $k$ and with no Berge Hamiltonian cycle. We also prove a stability theorem in the dense range before the first minimizer of $g_r(n,t)$: every near-extremal example is contained in one of two natural non-Hamiltonian constructions.

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arXiv2026

The $(t,p)$-Norm in Classical Extremal Problems

Given integers $r>t\ge1$ and a real number $p>0$, the $(t,p)$-norm $||\mathcal{H}||_{t,p}$ of an $r$-graph $\mathcal{H}$ is the sum of the $p$-th powers of the degrees $d_{\mathcal{H}}(T)$ over all $t$-subsets $T\subseteq V(\mathcal{H})$. When $t=r-1$, this is the codegree $p$-norm. For all sufficiently large $n$, we obtain the following results. The first two apply in both the convex range $p>1$ and the concave range $0 1$. In each of the three settings, we also characterize all extremal families.

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arXiv2026

Matchings and Near-Optimal 2-Factor Packings in Percolated Vertex-Transitive Graphs

Let $G$ be a connected simple vertex-transitive graph on $n$ vertices with degree $d$, and let $G_p$ be the random spanning subgraph obtained by retaining each edge of $G$ independently with probability $p$. Put $q:=1-p$. Motivated by a conjecture of Bedert, Dragani\'c, M\"uyesser, and Pavez-Sign\'e on Hamilton cycles in percolated Cayley graphs, we establish the corresponding matching and $2$-factor statements uniformly over the larger class of all connected vertex-transitive host graphs. For every $A>0$, if $q^d\le n^{-(5A+250)},$ then, with probability at least $1-n^{-A}$, the graph $G_p$ has a perfect matching when $n$ is even and is factor-critical when $n$ is odd. Separately, if $0<\epsilon<1$ and $ \epsilon^2pd\ge64(A+6)\log(2n), $ then, with probability at least $1-n^{-A}$, the graph $G_p$ contains at least \[ \left\lfloor\frac{(1-\epsilon)pd}{2}\right\rfloor \] pairwise edge-disjoint spanning $2$-factors. Moreover, if $pd/\log n\to\infty$, then \[ \nu_2(G_p)=(1+o(1))\frac{pd}{2} \] with high probability, which is asymptotically optimal, where $\nu_2(G)$ is the maximum number of pairwise edge-disjoint spanning 2-factors in $G$. Thus logarithmic-order percolation already forces these two factor-theoretic consequences of Hamiltonicity beyond the Cayley setting.

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arXiv2026

Counting Cycles in Graphs with Bounded Circumference

For an integer $L\ge2$, let $a=\lfloor L/2\rfloor$. Let $H(n,L)$ be the join of $K_a$ and an independent set of order $n-a$, with one extra edge in the independent set when $L$ is odd. We prove that, for fixed integers $q\ge4$ and $L>q$, and for all sufficiently large $n$, the graph $H(n,L)$ maximizes the number of copies of $C_q$ among all $n$-vertex graphs of circumference at most $L$. This settles a conjecture of Zhu, Gy\H{o}ri, He, Lv, Salia and Xiao~[Bull. Lond. Math. Soc. 55 (2023)]. For even $q\ge6$, we also prove the boundary case $L=q$. We further determine the corresponding maximum when a long path is forbidden.

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arXiv2026

Strong Subgraph-Count Stability in $C_{2\ell+1}$-Free Graphs

Starting from the stability theorem of Erd\H{o}s and Simonovits, stability problems for graphs forbidding a fixed subgraph have been studied in terms of edge numbers, spectral radii and subgraph counts. Let $\mathcal{N}(F,G)$ denote the number of unlabeled copies of $F$ in $G$. It is known that, for every fixed path $P_t$ and even cycle $C_{2a}$, the maximum number of copies in an $n$-vertex $C_{2\ell+1}$-free graph is attained by the bipartite Tur\'an graph $T_{n,2}$. In this paper we obtain strong structural stability for $C_{2\ell+1}$-free graphs in terms of copies of paths and even cycles. For fixed $\ell\ge2$ and $3\le r\le2\ell-1$, we show that if an $n$-vertex $C_{2\ell+1}$-free graph contains at least as many copies of $P_t$ or $C_{2a}$ as the corresponding suspended extremal construction, then it has the corresponding suspension structure. This gives exact high-chromatic extremal theorems for paths and even cycles. We also prove a counting theorem for nearly complete bipartite graphs. It shows that, for every fixed matching-admissible connected bipartite graph $F$, both imbalance between the two parts and missing cross-edges decrease the number of copies of $F$ by a term with a specified main coefficient. This theorem is independent of the forbidden odd cycle and converts subgraph-count assumptions into the edge bounds needed for the structural theorem.

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arXiv2026

Counting even cycles and even paths with bounded circumference

For an integer $L$, write $C_{\ge L}$ for the family of cycles of length at least $L$. For $L=2a$ let $H(n,L)=K_a+\overline K_{n-a}$, and for $L=2a+1$ let $H(n,L)$ be obtained from $K_a+\overline K_{n-a}$ by adding one edge inside the independent part. We prove sharp results for two even target graphs, namely even cycles $C_{2s}$ and even paths $P_{2r+1}$. For even cycles, with $s\ge3$ and $L\ge2s$, we have \[ \mathrm{ex}(n,C_{2s},C_{\ge L+1})=C_{2s}(H(n,L)) \] for all sufficiently large $n$. Together with the known $C_4$ case of Zhu, Gy\H{o}ri, He, Lv, Salia and Xiao~[Bull. Lond. Math. Soc. 55 (2023)], this verifies the even-cycle case of their conjecture on $\mathrm{ex}(n,C_k,C_{\ge L+1})$. For even paths, with $r\ge2$ and $L\ge2r$, we have \[ \mathrm{ex}(n,P_{2r+1},C_{\ge L+1})=N(P_{2r+1},H(n,L)) \] for all sufficiently large $n$. We also derive the corresponding exact results when the forbidden graph is a path $P_{p+1}$, sharpening the relevant even-cycle and even-path asymptotic results of Gy\H{o}ri, Salia, Tompkins and Zamora~[Discrete Math. Theor. Comput. Sci. 21 no. 1 (2019)].

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arXiv2026

Adjacent comparison bounds and extremal sets for Ruzsa numbers

Let $m$ be a positive integer and $\mathbb{Z}_m$ the residue class ring modulo $m$. The Ruzsa number $R_m$ is defined to be the least integer $r$ such that there is a subset $\mathcal{A}$ of $\mathbb{Z}_m$ satisfying $ 1\le \sigma_{\mathcal{A}}(n)\le r $ for any $n\in \mathbb{Z}_m$, where $$ \sigma_{\mathcal{A}}(n) =\#\big\{(a,a')\in\mathcal A^2: a+a'\equiv n\pmod{m}\big\}. $$ Motivated by a 2024 conjecture of Ding and Zhao, we prove $ | R_{m+1}-R_m|\le 144. $ Let $\mathcal{A}$ be a subset of $\mathbb{Z}_m$ satisfying $1\le \sigma_{\mathcal{A}}(n)\le R_m$ for any $n\in \mathbb{Z}_m$. We also give nontrivial bounds for the size of $\mathcal{A}$. Additionally, we provide exact values of $R_m$ for all $m\le 100$, which substantially extends the table of values given by S\'andor and Yang in 2017. Finally, we pose several related problems and prove some partial results.

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arXiv2026

On the Turán number of blow-ups of $\mathcal{F}_5$

Let $\mathcal{F}_5$ denote the $3$-uniform hypergraph on the vertex set $\{f_1,f_2,\dots,f_5\}$ with hyperedges $\{f_1f_2f_3,f_1f_2f_4,f_3f_4f_5\}$. Recently, Balogh, Clemen and Luo determined the Turán number of a one-vertex blow-up of $\mathcal{F}_5$, more specifically, they blow up the vertex $f_5$ to $t$ vertices, the resulting hypergraph is denoted by $\mathcal{F}_5(f_5;t)$. They show that for infinitely many $t$, $\mathcal{F}_5(f_5;t)$ has exponentially many extremal constructions and positive Turán density. In this paper, we determine the exact Turán number of the hypergraph obtained by blowing up $f_3$ of $\mathcal{F}_5$ to $t$ vertices and show that it also has exponentially many extremal constructions. We also give a general upper bound and lower bound of the Turán number of every blow-up of $\mathcal{F}_5$. For some special blow-ups of $\mathcal{F}_5$, for example, $t$-disjoint copies of $\mathcal{F}_5$, we determine the exact Turán number. We construct a hypergraph $\mathcal{F}_{sim}(t)$ which is a subgraph of a blow-up of $\mathcal{F}_5$, and is contained in the hypergraph obtained by adding any new hyperedge to the Turán hypergraph (the balanced complete $3$-partite hypergraph), but its extremal construction is not the Turán hypergraph. We also determine the exact Turán number of $\mathcal{F}_{sim}(t)$.

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arXiv2026

Hypergraph extensions of the Alon--Frankl Theorem and rainbow Turán problems

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 $F$. Recently, Alon and Frankl (JCTB, 2024) and Gerbner (JGT, 2023) studied the maximum number of edges in $n$-vertex $F$-free graphs with bounded matching number, respectively. Gerbner, Tompkins and Zhou (EJC, 2025) considered the analogous Turán problems on hypergraphs with bounded matching number. In this paper, we study hypergraph extensions of the Alon--Frankl Theorem. More precisely, we determine the maximum number of hyperedges in an $n$-vertex $r$-uniform hypergraph containing neither a matching $M^r_{s+1}$ nor the expansion $K_{\ell+1}^{(r)+}$ of the clique $K_{\ell+1}$ for all small $s<\frac{\ell^2-1}{2}$ and all sufficiently large $s$, respectively. This result partly confirms a conjecture proposed by Gerbner, Tompkins and Zhou (EJC, 2025). As a key tool, we determine the rainbow hyper-Turán number for expansions of cliques, which is defined as the maximum sum of size of a sequence of hypergraphs $\mathcal{H}_1,\dots,\mathcal{H}_k$ that contains no rainbow copies of expansions of cliques with given size. It extends the result of Keevash, Saks, Sudakov and Verstra{ë}te (AAM, 2004), which determined the rainbow Turán number of cliques in the graph case. These results shows a correlation between the hyper-Turán problem and the rainbow hyper-Turán number.

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arXiv2026

A generalization of Erdős-Hajnal problem on paths with equal-degree endpoints

Erdős and Hajnal proposed a problem that: is it true that every $(2n+1)$-vertex graph with $n^2+n+1$ edges contains two vertices of equal degree connected by a path of length three? The edge bound is sharp by the complete bipartite graph $K_{n,n+1}$. Recently, Chen and Ma [Journal of Combinatorial Theory, Series B, 179:1-18, 2026] answered this problem affirmatively for every $n \ge 600$. In the same paper, they further conjectured that for sufficiently large $n$, the statement is true if we replace the path of length three by a path of fixed odd length. In this paper, we confirm their conjecture.

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arXiv2026

On the connected Turán number of Berge paths and Berge cycles

Given a graph $F$, a Berge copy of $F$ (Berge-$F$ for short) is a hypergraph obtained by enlarging the edges arbitrarily. Győri, Salia and Zamora determined the maximum number of hyperedges in a connected $r$-uniform hypergraph on $n$ vertices containing no Berge path of length $k-1$ for all $k\geq 2r+14$ and sufficiently large $n$, and asked for the minimum $k_0$ such that this extremal number holds for all $k\geq k_0$. In this paper, we prove that the extremal number holds for all $k\geq 2r+2$ and fails for $k\le 2r+1$, thereby completely resolving the problem posed by Győri, Salia and Zamora. Moreover, we improve the result of Füredi, Kostochka and Luo, who determined the maximum number of hyperedges in a $2$-connected $n$-vertex $r$-uniform hypergraph containing no Berge cycle of length at least $k$ for all $k\geq 4r$ and sufficiently large $n$, by showing that this extremal number holds for all $k\geq 2r+2$ and fails for $k\le 2r+1$. Our approach reduces Berge-Turán problems to classical extremal graph theory problems, and applies recent work of Ai, Lei, Ning and Shi concerning the feasibility of graph parameters and the Kelmans operation.

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arXiv2026

Generalized Turán problems for Berge hypergraphs

Let $\mathcal{H}$ be a hypergraph and $F$ be a graph. If there exists a bijection between the hyperedges of $\mathcal{H}$ and the edges of $F$ such that each hyperedge contains its image, then we say that $\mathcal{H}$ is a \textit{Berge copy} of $F$, and the collection of Berge copies of $F$ is denoted by Berge-$F$. Given $r$-graphs $\mathcal{F}$ and $\mathcal{H}$, the generalized hyper-Turán number $\text{ex}_r(n, \mathcal{H}, \mathcal{F})$ is the maximum number of copies of $\mathcal{H}$ in $n$-vertex $\mathcal{F}$-free $r$-graphs. We study $\text{ex}_r(n, \mathcal{H}, \text{Berge-}F)$. For general $\mathcal{H}$, we connect this problem to counting copies of the shadow graph of $\mathcal{H}$ in $F$-free graphs and obtain several exact results. In particular, we show that for any hypergraph $\mathcal{H}$, if $k$ is sufficiently large, then $\text{ex}_r(n, \mathcal{H}, \text{Berge-}K_k)$ is achieved by the balanced complete $(k-1)$-partite $r$-graph, generalizing a result of Morrison, Nir, Norin, Rza{ż}ewski and Wesolek [\textit{Journal of Combinatorial Theory, Series B}, 162 (2023) 231--243] to the case of hypergraphs. We show that $\text{ex}_r(n,K_s^r,\text{Berge-}F)\le \text{ex}_s(n,\text{Berge-}F)$ and present sufficient conditions for equality. We also consider the connected generalized Turán number for Berge paths.

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arXiv2026

The Turán number of the Cartesian product of a star and an edge

Let $C_k$ denote the cycle of length $k$, $S_t$ be a star with $t$ edges. And let $B_t$ be the graph consisting of $t$ copies of $C_4$ sharing one fixed edge. Equivalently, $B_t=K_2 \mathbin{\square} S_t$, which is the Cartesian product of a star with $t$ edges and an edge. Recently, Gao, Janzer, Liu and Xu [\textit{Israel J. Math. 269(2025)}] proved that the Turán number of $K_2\mathbin{\square} C_{2l}$ is $Θ(n^{\frac{3}{2}})$ for every $l\ge 4$. In this paper, we obtain upper and lower estimates for the Turán number of $B_t$ in both the general and bipartite settings for every $t\geq 2$. For the lower bound, we use random construction based on the extremal structure of $C_4$. These results imply that $\frac{1}{2\sqrt{2}}\leq \lim_{t\to \infty} \frac{\mathrm{ex}(n,B_t)}{\sqrt{t}}\leq \frac{1}{2}$, and $\frac{1}{4}\leq \lim_{t\to \infty} \frac{\mathrm{ex}_{bip}(n,B_t)}{\sqrt{t}}\leq \frac{1}{2\sqrt{2}}.$ In the case of $B_2$, we obtain sharper estimates. We show that the Turán number of $B_2$ is approximately between $(0.518+o(1))n^{\frac{3}{2}}$ and $(0.603+o(1))n^{\frac{3}{2}}$. And in the bipartite setting, it is approximately between $(0.385+o(1))n^{\frac{3}{2}}$ and $(0.468+o(1))n^{\frac{3}{2}}$. Moreover, in the bipartite setting, we give a more general result, which shows that for every tree $T$ with $t$ edges, the bipartite Turán number of $K_2\mathbin{\square}T$ is at most $\frac{\sqrt{t}}{2\sqrt{2}}(1+o(1))n^{\frac{3}{2}}$.

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arXiv2026

Forbidding matching as trace in uniform hypergraphs

We say a hypergraph $\mathcal{H}$ contains a hypergraph $\mathcal{G}$ as trace if there exists a vertex subset $S \subseteq V(\mathcal{H})$ such that $|S| = |V(\mathcal{G})|$ and $\{e \cap S: e \in E(\mathcal{H})\}$ contains $\mathcal{G}$ as a sub-hypergraph. We use $\mathrm{ex}_r(n, \mathrm{Tr}_r(\mathcal{G}))$ to denote the maximum number of hyperedges in an $r$-uniform hypergraph on $n$ vertices not containing $\mathcal{G}$ as a trace. The study of Turán numbers for traces was initiated by Mubayi and Zhao who studied the case when $\mathcal{G}$ is a complete graph. Let $M_{s+1}$ denote the graph of a matching with $s+1$ edges. In this paper, we give the upper bound of $\mathrm{ex}_r(n, \mathrm{Tr}_r(M_{s+1}))$ which is sharp asymptotically. When $r=3$, we give the exact value of $\mathrm{ex}_3 (n, \mathrm{Tr}_3 (M_{s+1}))$. We also consider the generalized Turán number in the case of matching. That is, the maximum number of copies of clique $\mathcal{K}_t^r$ in hypergraphs forbidding $\mathrm{Tr}_r (M_{s+1})$ as a trace. We give an upper bound which is sharp asymptotically and when $r=3$, we give the exact value. The Turán number of forbidding a matching and the other graph is another well studied topic initiated by Alon and Frankl. We also consider an analogue problem for the trace version, i.e., forbidding trace of matching and trace of complete graph as subgraphs.

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arXiv2026

The number of induced paths in outerplanar graphs

Let $P_k$ denote the path with $k$ vertices, and $\mathrm{ex}_{\mathcal{OP}}(n,H^{\mathrm{ind}},\emptyset)$ be the maximum number of induced copies of $H$ in an $n$-vertex outerplanar graph. In this paper, we determine the exact value of $\mathrm{ex}_{\mathcal{OP}}(n,P_3^{\mathrm{ind}},\emptyset)$ for all $n$, and give an asymptotic value of $\mathrm{ex}_{\mathcal{OP}}(n,P_4^{\mathrm{ind}},\emptyset)$. For general $k$, Matolcsi and Nagy proved that $\lim_{k\to \infty} {\left( \mathrm{ex}_{\mathcal{OP}}(n, P_{k+1},\emptyset)\right)^{1/k}} =4$. In the induced case, we prove that \[ fib(k-1)\frac{{(n-2k+3)}^2}{4} \le \mathrm{ex}_{\mathcal{OP}}(n, P_{k+1}^{\mathrm{ind}},\emptyset) \le fib(k+1) \binom{n}{2}, \] where $fib(k)$ is the Fibonacci number. This implies that $\lim_{k\to \infty} {\left( \mathrm{ex}_{\mathcal{OP}}(n, P_{k+1}^{\mathrm{ind}},\emptyset)\right)^{1/k}} = \frac{\sqrt{5}+1}{2}\approx 1.618$.

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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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