arXiv subjects

Yaojun Chen

Publications and source records attributed to Yaojun Chen.

At least 19 recordsLinked to original sources

arXiv2026

Nonregular graphs of odd maximum degree with maximum spectral radius

Let $ρ(n,d)$ denote the maximum adjacency spectral radius among all connected nonregular graphs of order $n$ and maximum degree $d$. A graph attaining this maximum is called an extremal graph. Liu [J. Combin. Theory Ser. B, 2024] determined the extremal graphs for $d=3,4$ and formulated two conjectures for general $d$. For each fixed odd integer $d\ge3$, the conjectures assert that: (1) $\displaystyle\lim_{n\to\infty}n^2\bigl(d-ρ(n,d)\bigr) =(d-1)π^2/4$. (2) For all sufficiently large $n$, the degree sequence of every extremal graph is $(d,\ldots,d,d-1)$ for odd $n$ and $(d,\ldots,d,1)$ for even $n$. We prove the first conjecture for every fixed odd $d\ge3$ and, more precisely, obtain the asymptotic expansion \[ ρ(n,d) =d-\frac{(d-1)π^2}{4n^2} +\frac{(d-1)^2π^2}{4n^3} +O_d(n^{-4}) \qquad(n\to\infty). \] We further prove the second conjecture for every fixed odd $d\ge3$.

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arXiv2026

Ramsey numbers of sparse graphs versus disjoint books

Let $B_k$ denote a book on $k+2$ vertices and $tB_k$ be $t$ vertex-disjoint $B_k$'s. For any integers $k\ge2$ and $t\ge1$, there exists a positive constant $ε=ε(k,t)$ such that every connected graph $G$ with $n\ge111t^3k^3$ vertices and at most $n(1+ε)$ edges satisfies $$r(G,tB_k)=2n+t-2.$$ Our result extends the work of Erdős, Faudree, Rousseau, and Schelp (1988), who established the corresponding result for $G$ being a tree and $t=1$.

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arXiv2026

Fan-goodness of sparse graphs

Let $G$ be a connected graph of order $n$, $F_k$ be a fan consisting of $k$ triangles sharing a common vertex, and $tF_k$ be $t$ vertex-disjoint copies of $F_k$. Brennan (2017) showed the Ramsey number $r(G,F_k)=2n-1$ for $G$ being a unicyclic graph for $n \geq k^2-k+1$ and $k\ge 18$, and asked the threshold $c(n)$ for which $r(G,F_k) \geq 2n$ holds for any $G$ containing at least $c(n)$ cycles and $n$ being large. In this paper, we consider fan-goodness of general sparse graphs and show that if $G$ has at most $n(1+ε(k))$ edges, where $ε(k)$ is a constant depending on $k$, then $$r(G,F_k)=2n-1$$ for $n\ge 36k^4$, which implies that $c(n)$ is greater than $ε(k) n$. Moreover, if $G$ has at most $n(1+ε(k,t))$ edges, where $ε(k,t)$ is a constant depending on $k,t$, then $$r(G,tF_k)=2n+t-2$$ provided $n\ge 161t^2k^4$.

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arXiv2026

Ramsey multiplicity and extremal colorings for odd 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 Ramsey multiplicity $M(H,n)$ is the minimum number of monochromatic copies of $H$ over all red/blue edge-colorings of $K_n$. It is called threshold Ramsey multiplicity if $n=r(H)$, and denoted by $m(H)$. The only previously known general infinite family for which $m(H)$ has been determined is stars, due to Harary and Prins (1974). Let $C_k$ denote a cycle on $k$ vertices. Conlon, Fox, Sudakov, and Wei (2022) conjectured that $m(C_k)=(k-1)!/2$ for every sufficiently large odd integer $k$. In this paper, we determine $M(C_k,r(C_k)+\ell)$ for every fixed nonnegative integer $\ell$ and all sufficiently large odd $k$, and characterize all extremal colorings, thereby confirming the conjecture. This is also a second general infinite family for which $m(H)$ has been determined.

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arXiv2026

Minimum degree and sparse connected spanning subgraphs

Let $G$ be a connected graph on $n$ vertices and at most $n(1+ε)$ edges with bounded maximum degree, and $F$ a graph on $n$ vertices with minimum degree at least $n-k$, where $ε$ is a constant depending on $k$. In this paper, we prove that $F$ contains $G$ as a spanning subgraph provided $n\ge 6k^3$, by establishing tight bounds for the Ramsey number $r(G,K_{1,k})$, where $K_{1,k}$ is a star on $k+1$ vertices. Our result generalizes and refines the work of Erdős, Faudree, Rousseau, and Schelp (JCT-B, 1982), who established the corresponding result for $G$ being a tree. Moreover, the tight bound for $r(G,tK_{1,k})$ is also obtained.

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arXiv2026

A near-linear upper bound for Burr's conjecture

Let $f(k)$ denote the smallest integer such that every oriented graph $D$ with chromatic number at least $f(k)$ contains every oriented tree on $k$ vertices. Burr (1980) showed that $f(k)\le (k-1)^2$ and conjectured that $f(k)=2k-2$. Bessy, Gonçalves and Reinald (2025) proved that $f(k)=O(k^{3/2})$. In this paper, by using an absorbing set method, we show that $f(k)\le \lfloor 31\log (k!)\rfloor=O(k\log k)$.

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arXiv2026

Counterexamples to two conjectures on the diameter of clique-free graphs

Erdős et al. (JCT-B, 1989) conjectured that, for integers $r\ge 2$ and $δ\ge 2$ with $3r-1\midδ$, every connected $K_{2r+1}$-free graph of order $n$ and minimum degree $δ$ has diameter at most $ \frac{3r-1}{r}\cdot \frac{n}δ+O(1)$. Czabarka et al. (JCT-B, 2021) later proposed the following generalization: for every $k\ge 3$ and $δ\ge\left\lceil\frac{3k}{2}\right\rceil-1$, every connected $K_{k+1}$-free graph of order $n$ and minimum degree at least $δ$ has diameter at most $(3-\frac{2}{k})\cdot\frac{n}δ+O(1)$. We disprove the latter conjecture, including its $k$-colorable version, for every $k\ge 7$ and sufficiently large $δ$. When $k=2r\ge 8$ and $3r-1\midδ$, our construction also disproves the conjecture of Erdős et al. (JCT-B, 1989).

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arXiv2026

Any $k$-graph with zero $\ell$-degree Turán density is layered

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. Ding, Lamaison, Liu, Wang, and Yang (JLMS, 2025) studied the problem of what 3-graphs $F$ satisfy $π_{\mathrm{co}}(F) = 0$. They introduced layered $3$-graphs and conjectured that a $3$-graph has zero codegree Turán density if and only if it is layered and has zero uniform Turán density. For $k\ge 3$, a $k$-graph is called layered if its vertices can be labelled so that every edge has a unique maximum label and two edges with the same maximum label have the same label multiset. In this paper, we show that every non-layered $k$-graph $F$ on $m$ vertices satisfies \[ π_{\mathrm{co}}(F)\ge q_{k,m}^{-q_{k,m}}>0, \quad \text{where}\quad q_{k,m}=\frac{(k-1)^{m+1}-1}{k-2}, \] which implies any $k$-graph with zero $\ell$-degree Turán density is layered, and the case $k=3$ confirms the conjecture of Ding, Lamaison, Liu, Wang, and Yang.

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arXiv2026

New upper bound for the Ramsey number of odd cycles

The \emph{$k$-color Ramsey number} $R_k(C_{2\ell+1})$ is the least integer $n$ such that any $k$-edge-coloring of a complete graph $K_n$ has a monochromatic odd cycle $C_{2\ell+1}$. Axenovich, Cames van Batenburg, Janzer, Michel, and Rundström~(JCT-B, 2026) recently proved \[ R_k(C_{2\ell+1})\le (4\ell-2)^k k^{k/\ell}+1, \] and Miyazaki, Mulrenin, Pohoata, and Zheng further improved the factor $k^{k/\ell}$ to $(k!)^{1/\ell}$. As Jenssen and Skokan (AM, 2021) determined $R_k(C_{2\ell+1})$ for fixed $k$ and sufficiently large $\ell$, it becomes even more interesting to seek better bound for fixed $\ell$ and sufficiently large $k$. In this paper, we show \[ R_k(C_{2\ell+1}) \le \frac{2\ell}{2\ell-1}(2\ell-1)^k(k!)^{1/\ell} \exp\!\left(k^{1-1/\ell}+O_\ell\!\left(k^{1-2/\ell}+\log k\right)\right)+1 \] for every fixed $\ell\ge 2$ and sufficiently large $k$, which improves the bound of Miyazaki et al. by a factor $2^{k-o(k)}$, and the bound of Axenovich et al. by a factor $(2\e^{1/\ell})^{k-o(k)}$.

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arXiv2026

Erdős--Ko--Rado theorems in $\ell_2$-norm for three finite spaces

Let $\mathcal{F}$ be a $k$-uniform hypergraph. The famous Erdős--Ko--Rado (1961) theorem determines the maximum size and extremal structure for $\mathcal{F}$ being $t$-intersecting, that is, $|F_1 \cap F_2| \ge t$ for any two edges $F_1, F_2$ of $\mathcal F$. The codegree squared sum $\mathrm{co}_2(\mathcal{F})$ is the square of the $\ell_2$-norm of the codegree vector of all $(k-1)$-sets in $\mathcal{F}$, which was initially introduced for Turán problems of hypergraphs. Recently, Brooks and Linz (2026), as well as Wu and Zhang (2026) investigated the maximum value of $\mathrm{co}_2(\mathcal{F})$ and corresponding extremal structures for $\mathcal{F}$ being $t$-intersecting. Moreover, Brooks and Linz asked if the classical results on intersecting families can be extended to $\mathrm{co}_2(\mathcal{F})$. In this paper, by developing the spectral techniques for incidence matrices, we study the extremal problems of $\mathrm{co}_2(\mathcal{F})$ for $\mathcal{F}$ being intersecting families in finite vector spaces, affine spaces, and attenuated spaces, and establish the Erdős--Ko--Rado theorems in $\ell_2$-norm for the three finite spaces.

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arXiv2026

On monochromatic path covers conjecture of Erdős--Gyárfás

Erdős and Gyárfás conjectured in 1995 that, in every red--blue edge-coloring of a complete graph $K_n$, the vertex set can be covered by at most $\sqrt n$ monochromatic paths, all of the same color. Pokrovskiy, Versteegen and Williams (JCT-B, 2026) proved the conjecture for all sufficiently large $n$. In this paper, by using minimal counterexample method, we confirm the conjecture completely.

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arXiv2026

A note on tree-cycle Ramsey numbers

Let $R(T_n,C_m)$ denote the Ramsey number of a tree $T_n$ on $n$ vertices versus a cycle $C_m$ of length $m$. Burr, Erdős, Faudree, Rousseau, and Schelp (1982) asked for the least function $f(m)$ such that $R(T_n,C_m)=2n-1$ for every odd $m\ge 3$ whenever $n\ge f(m)$. They proved that $f(m)\le 756m^{10}$. This bound was later improved to $25m$ by Brennan (2016) and to $4m-8$ by Fan and Lin (2025). In this note, we show that $f(m)\le 2m-4$ by using a different method and conjecture that $f(m)=\lceil (2m-1)/3\rceil$.

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arXiv2026

Two-block cycles and chromatic number of Hamiltonian digraphs

Let $k$ and $\ell$ be positive integers. The family $C(k,\ell)$ consists of all digraphs obtained from two internally vertex-disjoint directed paths of lengths at least $k$ and $\ell$, respectively, and identifying their initial vertices and their terminal vertices. Addario-Berry, Havet and Thomassé (JCT-B, 2007) asked whether, for any positive integers $k$ and $\ell$ with $k+\ell \ge 4$, the chromatic number $χ(D)$ is at most $k+\ell-1$ for every $C(k,\ell)$-free strongly connected digraph $D$. Let $D$ be a $C(k,\ell)$-free Hamiltonian digraph. Kim, Kim, Ma and Park (JGT, 2018) showed that $χ(D) \le k+\ell$ and the bound is attained when $k+\ell=5$. In this paper, we prove that $χ(D) \le k+\ell-1$ for $k+\ell\ge 6$ and this bound is best possible for all $k+\ell\geq 6$, which resolves the problem posed by Addario-Berry, Havet and Thomassé for Hamiltonian digraphs.

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arXiv2026

On regular homogeneously traceable nonhamiltonian graphs

A graph is homogeneously traceable if each vertex is an endpoint of a Hamiltonian path. Chartrand, Gould, and Kapoor (1979) proved irregular homogeneously traceable nonhamiltonian graphs exist for every order $n\ge 9$. Hu and Zhan (DAM, 2022) considered the $3$-regular and $4$-regular cases and asked which order $n$ can be realized by a $k$-regular homogeneously traceable nonhamiltonian graph. Recently, Liu and Qiao (DAM, 2026) showed that $n=p(k-1)+q\ge 6(k-1)+q$ can be realized if $k\ge 5$ is odd and $q\in\{0,2,4,6\}$, or $k\ge 6$ is even and $q\in\{0,1,...,6\}$. In this paper, we show that for any $k\ge 6$ and $n\ge 6(k-2)$, there exists a $k$-regular homogeneously traceable nonhamiltonian graph of order $n$.

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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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