arXiv subjects

Bo Ning

Publications and source records attributed to Bo Ning.

At least 19 recordsLinked to original sources

arXiv2026

Every $r$-regular Hamiltonian graph with $r\geq 17$ contains a second Hamilton cycle

A conjecture of Sheehan states that every Hamiltonian $4$-regular graph contains a second Hamilton cycle. A natural problem is to find the smallest positive integer $k$ such that every Hamiltonian $r$-regular graph contains a second Hamilton cycle when $r\geq k$. Thomassen proved that $k\leq 300$. Improving Thomassen's result, Haxell, Seamone, and Verstra{ë}te proved that $k\leq 23$. We improve this bound by proving that every $r$-regular Hamiltonian graph contains a second Hamilton cycle whenever $r\geq 17$. Our new idea is to establish a transversal theorem for set systems on a cycle using Harris's results on the Moser--Tardos algorithm for Local Lemma. We then apply Thomassen's theorem on independent dominating sets to obtain a second Hamilton cycle. For $r\ge 18$, we also give some algorithms to find such a transversal.

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arXiv2026

A complete solution to the Boots-Royle/Cao-Vince conjecture

Boots and Royle, and independently Cao and Vince, conjectured that the join of an edge and a path on $n-2$ vertices is the unique planar graph of maximum adjacency spectral radius for $n\geq 9$. Tait and Tobin (JCTB, 2017) proved the conjecture for sufficiently large order. In this paper, we completely resolved the Boots-Royle/Cao-Vince conjecture.

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arXiv2026

A near-linear Chvátal--Erdős condition for Hamilton cycles in digraphs

For a digraph $D$, let $α_2(D)$ be the largest size of a vertex set containing no directed $2$-cycle. Let $f_2(a)$ be the least positive integer $k$ such that every $k$-strongly connected digraph $D$ with $α_2(D)\le a$ has a Hamilton cycle. Jackson and Ordaz conjectured that $f_2(a)\le a+1$. Towards this conjecture, we establish the near-linear bound $f_2(a)=O\!\left(\frac{a(\log a)^4}{(\log\log a)^2}\right),$ and hence $f_2(a)=O_\varepsilon(a^{1+\varepsilon})$ for every fixed $\varepsilon>0$. We also disprove the pancyclicity conjecture of Jackson and Ordaz that every digraph $D$ with $κ(D)\geα_2(D)+1$ contains a directed cycle of every length from $2$ to $|V(D)|$.

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arXiv2026

The spectral Erdős book theorem: sharp bounds and stability

The booksize $\mathrm{bk}(G)$ of a graph $G$ is the largest number of triangles sharing a common edge. A classical theorem of Edwards, conjectured by Bollobás and Erdős, states that every $n$-vertex graph $G$ with $e(G)>e(T_{n,2})$ has booksize greater than $n/6$. Zhai and Lin [J. Graph Theory 102 (2023) 502--520] asked whether the same conclusion holds under the spectral condition $λ(G)>λ(T_{n,2})$, where $λ(G)$ is the spectral radius of the adjacency matrix. We answer this question in a strong form: every $n$-vertex graph $G\neq T_{n,2}$ with $λ(G)\geλ(T_{n,2})$ satisfies \[ \mathrm{bk}(G)\ge\max\Big\{\frac13λ(G),\,λ(G)-\frac n3,\,2λ(G)-n\Big\}. \] Consequently, the condition $λ(G)>λ(T_{n,2})$ forces $\mathrm{bk}(G)\ge\lfloor n/6\rfloor+1$. The middle term is a spectral improvement of Edwards' bound $\mathrm{bk}(G)\ge\frac{2m}{n}-\frac n3$, and all three bounds are best possible. These results come from the edge-spectral setting: every graph $G$ with $m$ edges and $λ(G)\ge\sqrt m$ that is not a complete bipartite graph satisfies \[ \mathrm{bk}(G)\ge\max\Big\{λ(G)-\frac{2m}{3λ(G)},\,2λ(G)-\frac{2m}{λ(G)}\Big\}, \] which strengthens the bound $\mathrm{bk}(G)\ge\frac13λ(G)$ of Zhao, You, Zeng and Zhang. As an application of our method, we prove a triangle counting bound $t(G)\ge\frac13(λ(G)+1)(λ(G)^2-m)$, which improves the result of Bollobás and Nikiforov [J. Combin. Theory Ser B. (2007)]. Finally, we prove stability results at both thresholds: if $λ(G)\ge(\frac12-o(1))n$, then either $\mathrm{bk}(G)\ge(\frac16-o(1))n$ or $G$ can be made into $T_{n,2}$ by adding and deleting $o(n^2)$ edges; if $λ(G)\ge(1-o(1))\sqrt m$, then either $\mathrm{bk}(G)\ge(\frac13-o(1))\sqrt m$ or $G$ differs from a complete bipartite graph in $o(m)$ edges.

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arXiv2026

Longest cycles intersect linearly in highly connected graphs

A longstanding conjecture attributed to Smith (1984) asserts that for every $k\ge2$, any two longest cycles in a $k$-connected graph share at least $k$ vertices. In this paper, we prove the first linear lower bound, showing that any two longest cycles in a $k$-connected graph share at least $k/600$ vertices. Departing from previous Turán-type extremal arguments, we develop a novel structural approach that also yields applications to related problems on longest cycles and paths.

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arXiv2026

On Erdős Problem 767: Cycles with Chords

For integers $k\ge 1$ and $n\ge k+2$, let $g_k(n)$ be the maximum number of edges in an $n$-vertex graph containing no cycle with a vertex incident with at least $k$ chords. Erdős conjectured that $g_k(n)=(k+1)(n-k-1)$ for $n\ge 2k+2$. Lewin found a counterexample. Bollobás later conjectured that there exists a function $n(k)$ such that $g_k(n)=(k+1)(n-k-1)$ for all $n\ge n(k)$. Jiang confirmed this by proving the formula for all $n\ge3k+3$ when $k\ge1$. In this paper, we determine $g_k(n)$ completely. For all $k\ge1$ and $n\ge k+2$, we prove $g_k(n)=\big\{\lfloor\frac{(k+1)n}{2}\rfloor,\max\{a(n-a)+\lfloor\frac{a(k+1-a)}{2}\rfloor : a\in\mathbb Z,\; \lfloor {(k+1)}/{2}\rfloor+1\le a\le k+1\}\big\}$. For $k\ge2$, we prove $g_k(n)=(k+1)(n-k-1)$ when $n\ge \lceil(5k+1)/2\rceil$, and this threshold is sharp. Our proof builds on the method developed by Ma and the second author in [Ma and Ning, 2020].

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arXiv2026

Extensions of Erdős's 1962 theorem on non-Hamiltonian graphs

For a positive integer $k$, a graph property $\mathcal{H}$, and a graph parameter $\mathcal{P}$, let $\operatorname{ex}_{\mathcal{P}}(n, \mathcal{H}; δ\geq k)$ denote the maximum value of $\mathcal{P}$ over all $n$-vertex graphs with minimum degree at least $k$ that do not possess the property $\mathcal{H}$. The corresponding extremal families are denoted by $\operatorname{EX}_{\mathcal{P}}(n, \mathcal{H}; δ\geq k)$. For two disjoint graphs $H_1$ and $H_2$, let $H_1 \cup H_2$ denote their disjoint union, and let $H_1 \vee H_2$ denote their join. In 1962, Erdős established a classical theorem on the maximum number of edges in a non-Hamiltonian graph with prescribed order and minimum degree. Motivated by recent work on feasible graph parameters in \cite{ALNS2023}, we prove several extensions of Erdős's 1962 theorem on non-Hamiltonian graphs. The first result gives a common generalization of the extremal theorem due to Erdős and its spectral analogues. As direct applications, we obtain complete solutions to open problems raised in the literature since 2016, thereby improving nearly all related prior results in this direction.

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arXiv2026

On the difference between clique partition and clique covering numbers of graphs

For a graph $G$, let $\cpn(G)$ and $\ccn(G)$ denote the minimum numbers of cliques whose edge sets partition and cover $E(G)$, respectively, and put $f(n)=\max_{|V(G)|=n}\bigl(\cpn(G)-\ccn(G)\bigr).$ In 1983, Erdős, Faudree, and Ordman asked whether there is a sequence of graphs $G_n$ such that $|V(G_n)|=n$ and $\cpn(G_n)-\ccn(G_n)=n^2/4+O(n)$. The question appears as Problem 66 in Chung's survey \cite{ChungProblems} and is also listed on the UCSD Erdős Problems website. Caccetta, Erdős, Ordman, and Pullman proved that $f(n)=n^2/4-o(n^2)$. We prove that $f(n)=\left\lfloor\frac{n^2}{4}\right\rfloor-Θ(n^{4/3}),$ and hence answer the question in the negative.

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arXiv2026

Kohayakawa's conjecture and clique coverings of complements of paths and cycles

For $s\ge1$, let $G_s$ be the bipartite graph between the $s$-subsets and the $(s-1)$-subsets of $[2s]$, where adjacency means disjointness, and let $w(s)$ be the maximum number of $s$-subsets on an induced path in $G_s$. We prove $w(s)\ge \frac{4^s}{2048s^{5/2}}$ for all $s\geq 6$. This implies $\sup_{s\ge1}w(s)^{1/s}=4$, as conjectured by Kohayakawa (1991). His recursive construction then gives induced paths of order $Ω(4^r/r^{5/2})$ in the Kneser graph $KG(2r+1,r)$ and yields \[ \max\{\cc(\overline{P_n}),\ \cc(\overline{C_n})\} \le \log_2 n+\frac52\log_2\log_2 n+O(1). \] Together with the known lower bounds, this settles a conjecture of de Caen, Gregory, and Pullman (1985) and gives \[ \cc(\overline{P_n})=\log_2 n+Θ(\log_2\log_2 n), \qquad \cc(\overline{C_n})=\log_2 n+Θ(\log_2\log_2 n). \] We also give an independent proof of the latter order estimates. It uses a Hamiltonicity result of Kneser graphs and a key lemma proved by the Lovász local lemma.

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arXiv2026

On the difference between clique partition and clique covering numbers

For a graph $G$, let $\operatorname{cp}(G)$ and $\operatorname{cc}(G)$ be the minimum numbers of cliques in an edge partition and a clique cover of $G$, respectively. Set $σ_n = \max_{\lvert V(G)\rvert=n} \bigl(\operatorname{cp}(G)-\operatorname{cc}(G)\bigr), d_n=\left\lfloor\frac{n^2}{4}\right\rfloor-σ_n.$ In 1983, Erdős, Faudree, and Ordman asked whether $d_n=O(n)$. Caccetta, Erdős, Ordman, and Pullman previously constructed graphs showing $d_n=O(n^{3/2})$. We prove that $d_n=Θ(n^{4/3}),$ thereby determining the correct order of the deficit and answering their question in the negative.

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arXiv2026

Nikiforov's spectral consecutive cycle problem and the connected-matching method

Let $ρ(G)$ denote the adjacency spectral radius of a graph $G$ of order $n$. We determine the sharp constant in an open problem of Nikiforov (2008) on cycles of consecutive lengths. For every $\varepsilon>0$ and all sufficiently large $n$, if $G$ is an $n$-vertex graph with $ρ(G)>\sqrt{\lfloor{n^2/4}\rfloor},$ then $G$ contains a cycle $C_\ell$ for every integer length $3\le \ell\le (\frac{3-\sqrt5}{2}-\varepsilon)n.$ The constant $(3-\sqrt5)/2$ is best possible, as shown by the split graph $K_k\vee\overline K_{n-k}$ with $k\sim(3-\sqrt5)n/4$. Our result improves all previous results [LAA2008, CPC2020, JGT2023, JGT2023, GC2024]. The proof combines the degree form of Szemerédi's regularity lemma, a spectral matching theorem of Feng-Yu-Zhang, Weyl's inequality, a refinement of Łuczak's connected-matching embedding method, and other ideas.

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arXiv2026

Cycle lengths and chords under chromatic and degree constraints

We mainly consider three problems on cycle lengths and cycles with chords in graphs: (a) Gao, Huo, and Ma \cite[Question~1.5]{GaoHuoMa2021} asked whether, for every fixed $k\ge3$, there is a function $f_k(n)\to\infty$ such that every $n$-vertex $(k+1)$-critical graph contains $f_k(n)$ consecutive cycle lengths. (b) Let $g_k(n)$ be the maximum integer $t$ such that every $n$-vertex $k$-critical graph with $k\ge4$ contains an odd cycle with at least $t$ chords. Voss conjectured (see \cite[pp.~168]{VossBook}) that $g_k(n)\to\infty$ as $n\to\infty$ for each $k\ge4$, which extends a 1976 conjecture of Erdős (see also Erdős Problem~1091 \cite{Bloom1091}). (c) Kára and Král \cite{KaraKral2003} conjectured that every graph on $31$ vertices with minimum degree at least $8$ contains a cycle with at least $31$ chords. We answer question (a) in the negative for $k=3$, and disprove conjecture (b) for all $k\ge5$. We point out the work of Alexeev-Putterman-Sawhney-Sellke-Valiant (2026) on Erdős Problem 1901 disproves the case $k=4$ for conjecture (b). We prove conjecture (c). We also discuss two other related problems in the part of concluding remark.

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arXiv2026

The largest Laplacian eigenvalue of induced-$K_{1,r}$-free graphs

Let $G$ be a simple graph of maximum degree $d$, and let $μ(G)$ denote the largest eigenvalue of its Laplacian matrix. For a fixed integer $k\geq 2$, Aharoni, Alon, and Berger (2016) asked whether every graph containing no induced copy of $K_{1,k}$ satisfies $μ(G)\leq (2 - \frac{2}{k} + o(1)) d$. We answer this question by proving the stronger sharp bound \[ μ(G)\leq \left(2-\frac{2}{k}\right)(d+1). \] The proof combines a sign decomposition of a Laplacian Rayleigh vector with a weighted local Caro-Wei type inequality for independent sets.

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arXiv2026

Two problems of Burr, Erd\H os, Graham, and Sós on maximal anti-Ramsey functions for $P_4$

Burr, Erd\H os, Graham, and Sós introduced the maximal anti-Ramsey function $χ_{\mathrm{S}}(n,e,L)$, the minimum number of colors required over all $n$-vertex graphs with at least $e$ edges such that every copy of $L$ is rainbow. In \cite{BEGS1989}, they posed the following two problems: (i) Is it true that there exists $C>0$, such that for all $u\ge 1$, $χ_{\mathrm{S}}\left(n,\lfloor un \rfloor,P_4 \right) 0$, there exists $c(ε)>0$ such that for all sufficiently large $n$, \\ $χ_{\mathrm{S}}\left(n,\binom{n}{2}-\lfloor n^{2-ε} \rfloor,P_4 \right)>c(ε)n^{2}$? In this note, we give an affirmative answer to the first problem and a negative answer to the second problem. For the first problem, our proof uses a local density inequality with strong edge-colorings of odd Kneser graphs. In particular, our proof uses the characterization by Lužar, Máčajová, Škoviera, and Soták of~$k$-regular graphs whose strong chromatic index equals~$2k-1$. For the second result, our main tool is the construction of Alon, Moitra, and Sudakov. We show that for every fixed~$0<ε<1/2$ there exist~$γ>0$ and arbitrarily large~$n$ such that~$χ_{\mathrm{S}}\bigl(n,\tbinom{n}{2}-\lfloor n^{2-ε}\rfloor,P_4\bigr)\;\le\; n^{2-γ}=o(n^{2}).$

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arXiv2026

Rainbow triangles in edge-colored graphs with large minimum color degree

Let $G$ be an edge-colored graph on $n$ vertices, and let $\deltac(G)$ denote its minimum color degree. Li and, independently Li, Ning, Xu, and Zhang, proved that every edge-colored graph on $n$ vertices with $\deltac(G) \ge \frac{n+1}{2}$ contains a rainbow triangle. Let $\rt(G)$ denote the number of rainbow triangles in $G$, and define \[ f(n) = \min\{ \rt(G) : |V(G)| = n,\ \deltac(G) \ge (n+1)/2 \}. \] In \cite{LiNingShiZhang2024}, the following open problem was posed: determine all the values of $f(n)$. In this paper, we determine $f(n)$ completely: $f(n) = (n^2-1)/8$ for odd $n\geq 3$, $f(n) = \frac{n^2}{4} - 1$ for all even $n \ge 6,$ and $f(4) = 4$. This resolves an open problem raised in \cite{LiNingShiZhang2024}.

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arXiv2026

Counting triangles in graphs with no wheels of order at least five

For a family of graphs $\mathcal F$, a graph $G$ is said to be $\mathcal F$-free if it contains no member of $\mathcal F$ as a subgraph. A wheel graph $W_k$ is a graph on $k+1$ vertices formed by joining a new vertex to all vertices of a $k$-cycle. Given an integer $k\ge 3$, we consider the problem of determining the maximum number of triangles in a $W_{\geq k}$-free graph, where $W_{\geq k}=\{W_\ell: \ell \geq k\}$. The case $k=3$ was raised by Gallai, who proposed a conjecture for this case (see Erdős [5]. Gallai's conjecture was disproved by Zhou [17] and independently by Füredi, Goemans, and Kleitman [9]. In this paper, we study the case $k=4$. Namely, for every integer $n\ge 3$, we determine the maximum number of triangles in an $n$-vertex $W_{\geq 4}$-free graph and characterize all extremal graphs.

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arXiv2026

Longest cycles and Dirac-type results in highly connected graphs

A classical theorem of Nash-Williams states that if $G$ is a $2$-connected graph on $n$ vertices with minimum degree at least $(n+2)/3$, then for every longest cycle $C$ of $G$, the graph $G-V(C)$ is edgeless. Motivated by a higher-connectivity analogue, Bondy conjectured in 1980 that if $G$ is a $k$-connected graph on $n$ vertices with minimum degree at least $(n+k(k-1))/(k+1)$, then for every longest cycle $C$ of $G$, every path in $G-V(C)$ has at most $k-1$ vertices. This conjecture is known for $k\le 3$ and remains open for all $k\ge 4$. In this paper, we prove Bondy's conjecture for all sufficiently large graphs. The key ingredient is a new Dirac-type theorem that gives a lower bound on the length of a longest cycle in a $k$-connected graph, which also yields a partial solution to a conjecture of Jung from 1990. Along the way, we develop several new tools, including a DFS lemma and an average-degree analogue of the Bondy--Jackson theorem. We conclude with a discussion of related problems and a counterexample to a conjecture of Voss from 1991.

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arXiv2026

A Proof of a Conjecture on Positive and Negative Square Energies of Unicyclic Graphs

Let $G$ be a unicyclic graph of order $n$, and let $k$ be the length of the unique cycle of $G$. For the adjacency eigenvalues of $G$, let $s^{+}(G)$ and $s^{-}(G)$ denote the sums of the squares of the positive and negative eigenvalues, respectively. Akbari, Kumar, Mohar, Pragada, and Zhang conjectured that, when $k$ is odd, the value of $k$ modulo $4$ determines which of $s^+(G)$ and $s^-(G)$ is greater than $n$. More precisely, if $k\equiv 3\pmod 4$, then $s^+(G)>n>s^-(G)$; if $k\equiv 1\pmod 4$, then $s^+(G)<n<s^-(G)$. We confirm this conjecture.

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