arXiv subjects

Yaojun Chen

Publications and source records attributed to Yaojun Chen.

At least 37 records · Page 2Linked to original sources

arXiv2026

On the codegree Turán density of projective geometries

Let $F$ be a $k$-uniform hypergraph, abbreviated as $k$-graph. The codegree Turán density $γ(F)$ is the supremum over all $γ\in [0,1)$ such that, for arbitrarily large $n$, there exists an $n$-vertex $F$-free $k$-graph $H$ whose every $(k-1)$-subset of vertices lies in at least $γn$ edges. Let $PG_m(q)$ be the projective geometry of dimension $m$ over finite field $\mathbb{F}_q$. In this paper, we prove that $γ(PG_m(q)) \ge \frac{1}{p}> 0$ for all $m$ and $q$, where $p$ is the smallest prime divisor of $q+1$. This resolves an open problem proposed by Keevash and Zhao (JCT-B, 2007). Moreover, we determine the exact codegree Turán density of $PG_4(q)$ when $q$ is an odd prime power.

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arXiv2026

Odd covers for complete graphs and complete 3-graphs

The Graham-Pollak theorem says that one needs at least $n - 1$ complete bipartite graphs to cover each edge of a complete graph $K_{n}$ on $n$ vertices exactly once. The odd cover problem is a parity analogue which seeks the minimum number of complete bipartite graphs, denoted by $b_2(n)$, such that each edge of $ K_n $ is covered an odd number of times. An odd cover of a complte 3-graph $K_n^{(3)}$ on $n$ vertices is a family of complete $3$-partite $3$-graphs such that every triple is covered an odd number of times. Let $b_3(n)$ be the minimum size of such a family. The values of $b_2(n)$ and $b_3(n)$ are determined for some $n$ in several previous works. In this paper, we first determine the value of $b_2(n)$ for all $n$, which confirms a conjecture due to Buchanan et al. (JGT, 2026), and then show $b_3(n+1)=b_2(n)$ by which the value of $b_3(n)$ is determined for all $n$, that resolves a question posed by Leader and Tan (EJC, 2026).

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arXiv2026

On linear $k$-graphs with codegree Turán density arbitrarily close to zero

Let $F$ be a $k$-uniform hypergraph, abbreviated as $k$-graph. The codegree Turán density $π_{\mathrm{co}}(F)$ is the supremum over all $γ\in [0,1)$ such that, for arbitrarily large $n$, there exists an $n$-vertex $F$-free $k$-graph $H$ whose every $(k-1)$-subset of vertices lies in at least $γn$ edges. In this paper, we prove that there is a linear $k$-graph $F$ with $0<π_{co}(F) < \varepsilon$ for any $\varepsilon>0$. The special case $k=3$ solve a question proposed by Ding, Lamaison, Liu, Wang and Yang (JLMS, 2025). The main method combines an affine-plane-type incidence structure over a finite field and elementary number-theoretic arguments.

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arXiv2026

On the threshold Ramsey multiplicity conjectures for paths and even cycles

The Ramsey number $r(H)$ of a graph $H$ is the minimum positive integer $n$ such that every red/blue edge-coloring of the complete graph $K_n$ on $n$ vertices contains a monochromatic copy of $H$. The threshold Ramsey multiplicity $m(H)$ of $H$ is the minimum number of monochromatic copies of $H$ over all red/blue edge-colorings of $K_{r(H)}$. Let $P_t$ and $C_t$ be a path and a cycle on $t$ vertices, respectively. In this paper, by using combinatorial and local random construction, we show that $$m(C_{2t})\le t^{-γ+o(1)}\frac{(2t-1)!}{2}, \qquad m(P_{2t+1})\le t^{-γ+o(1)}\frac{t}{2}(2t)!,$$ and $$m(P_{2t})\leq \left(\frac{7}{8}+o(1)\right)\frac{(2t)!}{2},$$ for sufficiently large $t$, where $γ=1/(1+\sqrt{2})$. These results disprove two conjectures on the threshold Ramsey multiplicity for even cycles and paths, due to Conlon, Fox, Sudakov, and Wei.

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arXiv2026

Ramsey numbers and Gallai--Ramsey numbers of disjoint unions of cherries

For graphs $G_1,\ldots,G_k$, the Ramsey number $R(G_1,\ldots,G_k)$ is the smallest positive integer $N$ such that every $k$-edge-coloring of $K_N$ contains a monochromatic copy of $G_i$ in color $i$ for some $i\in[k]$. The Gallai--Ramsey number $GR(G_1,\ldots,G_k)$ is defined analogously, with the colorings restricted to Gallai colorings (i.e., edge-colorings with no rainbow triangle). A copy of $P_3$ is called a cherry. Let $n_iP_3$ denote the disjoint union of $n_i$ cherries. Wu, Magnant, Nowbandegani, and Xia (Discrete Appl. Math., 2019) proposed two conjectures: \[ R(n_1P_3,\ldots,n_kP_3)=N\ \text{and}\ GR(n_1P_3,\ldots,n_kP_3)=N\,, \] where $N=2\max\{n_1,\ldots,n_k\}+\sum_{i=1}^kn_i-k+1$. We disprove the Ramsey conjecture and provide some sufficient conditions for determining the exact value of $R(n_1P_3,\ldots,n_kP_3)$. In contrast, we confirm the Gallai--Ramsey conjecture.

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arXiv2026

New bounds for Ramsey numbers involving graphs with a center

Let $F_n$, $W_n$, and $\widehat{K}_n$ be the graphs obtained by joining a vertex to $n$ independent edges, a cycle and a path of order $n-1$, respectively. In this paper, we give new bounds for the Ramsey numbers $R(F_n,F_m)$ and $R(W_n,W_n)$, which improve those due to Chen, Yu, and Zhao [EJC, 2021] and Mao, Wang, Magnant, and Schiermeyer [G&C, 2022], respectively, and establish lower and upper bounds for $R(\widehat{K}_n,\widehat{K}_n)$. Moreover, we present a blow-up technique to establish some new lower bounds for the Ramsey numbers of wheels versus cliques.

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arXiv2026

The Turán number of the triangular pyramid of 4-layers

The Turán number $ex(n,H)$ of a graph $H$ is the maximum number of edges in any $H$-free graph on $n$ vertices. The triangular pyramid of $k$-layers, denoted by $TP_k$, is a generalization of a triangle. The Turán problems of a triangular pyramid with small layers have been studied widely by Liu (E-JC, 2013), Xiao, Katona, Xiao and Zamora (DAM, 2022), Ghosh, Győri, Paulos, Xiao and Zamora (DAM, 2022). Moreover, Ghosh et al. conjectured that $ex(n, TP_4)=\frac{1}{4}n^2+Θ(n^{\frac{4}{3}})$. In this note, we confirm this conjecture.

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arXiv2025

Trichotomy and $tK_m$-goodness of sparse graphs

Let $G$ be a connected graph with $n$ vertices and $n+k-2$ edges and $tK_m$ denote the disjoint union of $t$ complete graphs $K_m$. In this paper, by developing a trichotomy for sparse graphs, we show that for given integers $m\ge 2$ and $t\ge 1$, there exists a positive constant $c$ such that if $1\le k\le cn^{\frac{2}{m-1}}$ and $n$ is large, then $G$ is $tK_m$-good, that is, the Ramsey number is \[ r(G, tK_m)=(n-1)(m-1)+t\,. \] In particular, the above equality holds for any positive integers $k$, $m$, and $t$, provided $n$ is large. The case $t=1$ was obtained by Burr, Erdős, Faudree, Rousseau, and Schelp (1980), and the case $k=1$ was established by Luo and Peng (2023).

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arXiv2025

Oriented diameter of graphs with given domination number

Let $G$ be a connected bridgeless graph with domination number $γ$. The oriented diameter (strong diameter) of $G$ is the smallest integer $d$ for which $G$ admits a strong orientation with diameter (strong diameter) $d$. Kurz and Lätsch (2012) conjectured the oriented diameter of $G$ is at most $\lceil \frac{7γ+1}{2}\rceil$ and the bound is sharp. In this paper, we confirm the conjecture by induction on $γ$ through contracting an unavoidable alternative subgraph, which holds potential for future applications. Moreover, we show the oriented strong diameter of $G$ is at most $7γ-1$ by using the same recursive structure, and the bound is best possible.

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arXiv2025

Oriented Ramsey numbers of some sparse graphs

Let $H$ be an oriented graph without directed cycle. The oriented Ramsey number of $H$, denoted by $\overrightarrow{r}(H)$, is the smallest integer $N$ such that every tournament on $N$ vertices contains a copy of $H$. Rosenfeld (JCT-B, 1974) conjectured that $\overrightarrow{r}(H)=|H|$ if $H$ is a cycle of sufficiently large order, which was confirmed for $|H|\geq 9$ by Zein recently, and so does if $H$ is a path. Note that $\overrightarrow{r}(H)=|H|$ implies any tournament contains $H$ as a spanning subdigraph, it is interesting to ask when $\overrightarrow{r}(H)=|H|$ for $H$ being a sparse oriented graph. Sós (1986) conjectured this is true if $H$ is a directed path plus an additional edge containing the origin of the path as one end, which was confirmed by Petrović (JGT, 1988). In this paper, we show that $\overrightarrow{r}(H)=|H|$ for $H$ being an oriented graph obtained by identifying a vertex of an antidirected cycle with one end of a directed path. Some other oriented Ramsey numbers for oriented graphs with one cycle are also discussed.

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arXiv2024

The Turán number of path-star forests

The Turán number of a graph $H$, denoted by $ex(n,H)$, is the maximum number of edges in any graph on $n$ vertices containing no $H$ as a subgraph. A linear (star) forest is a forest consisting of paths (stars). A path-star forest $F$ is a forest consisting of paths and stars. In this paper, we determine $ex(n,F)$ for sufficiently large $n$ and characterize the corresponding extremal graphs, and our result generalizes previous known results on the Turán numbers of linear forests and star forests.

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arXiv2024

Disproofs of four Gallai-Ramsey-type conjectures

As a significant variation of Ramsey numbers, the Gallai-Ramsey number $GR_k(H)$ refers to the smallest positive integer $r$ such that, by coloring the edges of $K_r$ with at most $k$ colors, there exists either a monochromatic subgraph isomorphic to $H$ or a rainbow triangle. Mao, Wang, Magnant, and Schiermeyer [Discrete Math., 2023], Song, Wei, Zhang, and Zhao [Discrete Math., 2020], and Zhao and Wei [Discrete Appl. Math., 2021] each proposed one conjecture on the Gallai-Ramsey numbers for fans, wheels, and kipases, respectively. We establish new lower bounds that disprove all three conjectures. Su and Liu [Graphs Combin., 2022] studied the Gallai-Ramsey-full property of graphs and conjectured that a graph is Ramsey-full if and only if it is Gallai-Ramsey-full. We present two classes of graphs that are Ramsey-full, but neither is Gallai-Ramsey-full.

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arXiv2024

Generalized Turán problem for a path and a clique

Let $\mathcal{H}$ be a family of graphs. The generalized Turán number $ex(n, K_r, \mathcal{H})$ is the maximum number of copies of the clique $K_r$ in any $n$-vertex $\mathcal{H}$-free graph. In this paper, we determine the value of $ex(n, K_r, \{P_k, K_m \} )$ for sufficiently large $n$ with an exceptional case, and characterize all corresponding extremal graphs, which generalizes and strengthens the results of Katona and Xiao [EJC, 2024] on $ex(n, K_2, \{P_k, K_m \} )$. For the exceptional case, we obtain a tight upper bound for $ex(n, K_r, \{P_k, K_m \} )$ that confirms a conjecture on $ex(n, K_2, \{P_k, K_m \} )$ posed by Katona and Xiao.

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arXiv2024

Inverse model for network construction: (δ(G), I' (G)) -> G

The isolated toughness variant is a salient parameter for measuring the vulnerability of networks, which is inherently related to fractional factors (used to characterize the feasibility of data transmission). The combination of minimum degree and the corresponding tight bound of isolated toughness variant for fractional factors provide reference standards for network construction. However, previous advances only focused on how to select the optimal parameter criteria from Pareto front, without any suggestion for the construction of specific networks. To overcome this deficiency, this paper proposes an inverse model from $(δ(G),I'(G))$ to $G$, by means of evolutionary computing approach, we propose a novel inverse model to obtain the optimal solutions for candidate graphs. The main procedure is composed of pseudo-greedy acceleration, cross-mutation and diversity enhancement modules. The practicality of the algorithm is verified by means of pilot experiments. The code in this paper is made public on https://github.com/AizhEngHN/Inverse-model-for-network-construction-G-I-G-G.

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arXiv2023

Ramsey goodness of fans

Given two graphs $G_1$ and $G_2$, the Ramsey number $r(G_1,G_2)$ refers to the smallest positive integer $N$ such that any graph $G$ with $N$ vertices contains $G_1$ as a subgraph, or the complement of $G$ contains $G_2$ as a subgraph. A connected graph $H$ is said to be $p$-good if $r(K_p,H)=(p-1)(|H|-1)+1$. A generalized fan, denoted as $K_1+nH$, is formed by the disjoint union of $n$ copies of $H$ along with an additional vertex that is connected to each vertex of $nH$. Recently Chung and Lin proved that $K_1+nH$ is $p$-good for $n\ge cp\ell/|H|$, where $c\approx 52.456$ and $\ell=r(K_{p},H)$. They also posed the question of improving the lower bound of $n$ further so that $K_1+nH$ remains $p$-good. In this paper, we present three different methods to improve the range of $n$. First, we apply the Andrásfai-Erdős-Sós theorem to reduce $c$ from $52.456$ to $3$. Second, we utilize the approach established by Chen and Zhang to achieve a further reduction of $c$ to $2$. Lastly, we employ a new method to bring $c$ down to $1$. In addition, when $K_1+nH$ forms a fan graph $F_n$, we can further obtain a slightly more refined bound of $n$.

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arXiv2023

Maximum cliques in a graph without disjoint given subgraph

The generalized Turán number $\ex(n,K_s,F)$ denotes the maximum number of copies of $K_s$ in an $n$-vertex $F$-free graph. Let $kF$ denote $k$ disjoint copies of $F$. Gerbner, Methuku and Vizer [DM, 2019, 3130-3141] gave a lower bound for $\ex(n,K_3,2C_5)$ and obtained the magnitude of $\ex(n, K_s, kK_r)$. In this paper, we determine the exact value of $\ex(n,K_3,2C_5)$ and described the unique extremal graph for large $n$. Moreover, we also determine the exact value of $\ex(n,K_r,(k+1)K_r)$ which generalizes some known results.

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arXiv2023

Extremal problems for a matching and any other graph

For a family of graphs $\F$, a graph is called $\F$-free if it does not contain any member of $\F$ as a subgraph. The generalized Turán number $\ex(n,K_r,\F)$ is the maximum number of $K_r$ in an $n$-vertex $\F$-free graph and $\ex(n,K_2,\F)=\ex(n,\F)$, i.e., the classical Turán number. Let $M_{s+1}$ be a matching on $s+1$ edges and $F$ be any graph. In this paper, we determine $\ex(n,K_r, \{M_{s+1},F\})$ apart from a constant additive term and also give a condition when the error constant term can be determined. In particular, we give the exact value of $\ex(n,\{M_{s+1},F\})$ for $F$ being any non-bipartite graph or some bipartite graphs. Furthermore, we determine $\ex(n,K_r,\{M_{s+1},F\})$ when $F$ is color critical with $χ(F)\ge \max\{r+1,4\}$. These extend the results in [2,11,18].

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arXiv2023

The hat guessing number of random graphs with constant edge-chosen probability

Let $G$ be a graph with $n$ vertices. The {\em hat guessing number} of $G$ is defined in terms of the following game: There are $n$ players and one opponent. The opponent will wear one of the $q$ hats of different colors on the player's head. At this time, the player can only see the player's hat color at the adjacent vertex, and communication between players is not allowed. Once players are assigned hats, each player must guess the color of his hat at the same time. If at least one player guesses right, then they will win collectively. Given a graph $G$, its hat guessing number $HG(G)$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ different colors. Let $\mathcal{G}(n,p)$ denote the Erdős-Rényi random graphs with $n$ vertices and edge-chosen probability $ p\in(0,1)$. Alon-Chizewer and Bosek-Dudek-Farnik-Grytczuk-Mazur investigated the lower and upper bound for $HG(G)$ when $G\in \mathcal{G}(n,1/2)$, respectively. In this paper, we extends their results by showing that for any constant number $p$, we have $n^{1 - o(1)}\le HG(G) \le (1-o(1))n$ with high probability when $G\in \mathcal{G}(n,p)$.

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