arXiv subjects

Bo Ning

Publications and source records attributed to Bo Ning.

At least 19 recordsLinked to original sources

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, Erd\H{o}s, 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 Erd\H{o}s Problems website. Caccetta, Erd\H{o}s, Ordman, and Pullman proved that $f(n)=n^2/4-o(n^2)$. We prove that $f(n)=\left\lfloor\frac{n^2}{4}\right\rfloor-\Theta(n^{4/3}),$ 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 $\Omega(4^r/r^{5/2})$ 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 \[ \cc(\overline{P_n})=\log_2 n+\Theta(\log_2\log_2 n), \qquad \cc(\overline{C_n})=\log_2 n+\Theta(\log_2\log_2 n). \] 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 Lov\'{a}sz local lemma.

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arXiv2026

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

Let $\rho(G)$ 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 $\rho(G)>\sqrt{\lfloor{n^2/4}\rfloor},$ 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 Szemer\'edi's regularity lemma, a spectral matching theorem of Feng-Yu-Zhang, Weyl's inequality, a refinement of \L{}uczak's connected-matching embedding method, and other ideas.

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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

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 Erd\H{o}s (see also Erd\H{o}s Problem~1091 \cite{Bloom1091}). (c) K\'ara and Kr\'al \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 Erd\H{o}s 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 $\mu(G)$ 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 $\mu(G)\leq (2 - \frac{2}{k} + o(1)) d$. We answer this question by proving the stronger sharp bound \[ \mu(G)\leq \left(2-\frac{2}{k}\right)(d+1). \] 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 S\'os on maximal anti-Ramsey functions for $P_4$

Burr, Erd\H os, Graham, and S\'os introduced the maximal anti-Ramsey function $\chi_{\mathrm{S}}(n,e,L)$, 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 $u\ge 1$, $\chi_{\mathrm{S}}\left(n,\lfloor un \rfloor,P_4 \right) 0$, there exists $c(\epsilon)>0$ such that for all sufficiently large $n$, \\ $\chi_{\mathrm{S}}\left(n,\binom{n}{2}-\lfloor n^{2-\epsilon} \rfloor,P_4 \right)>c(\epsilon)n^{2}$? 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 Lu\v{z}ar, M\'{a}\v{c}ajov\'a, \v{S}koviera, and Sot\'ak 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~$0<\epsilon<1/2$ there exist~$\gamma>0$ and arbitrarily large~$n$ such that~$\chi_{\mathrm{S}}\bigl(n,\tbinom{n}{2}-\lfloor n^{2-\epsilon}\rfloor,P_4\bigr)\;\le\; n^{2-\gamma}=o(n^{2}).$

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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 Erd\H{o}s [5]. Gallai's conjecture was disproved by Zhou [17] and independently by F\"uredi, 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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arXiv2026

On spectral Tur\'an theorems: confirming a conjecture of Guiduli and two problems of Nikiforov

Let $G$ be an $n$-vertex graph, and let $\lambda(G)$ and $\lambda_n(G)$ denote the largest and smallest eigenvalues of its adjacency matrix. Write $e(G)$ for the number of edges of $G$, $d(G)=2e(G)/n$ for its average degree, and $T_r(n)$ for the $r$-partite Tur\'an graph on $n$ vertices. We prove four sharp results in spectral Tur\'an theory. First, we confirm Guiduli's spectral dense-neighborhood conjecture (1996) in a stronger form: if $\lambda(G)\ge \lambda(T_r(n))$, then either $G\cong T_r(n)$, or there exists a vertex $v$ such that $\lambda(G[N(v)]) > \lambda(T_{r-1}(d(v)))$. Moreover, when $\lambda(G)>\lambda(T_r(n))$, every vertex attaining the maximum entry in any nonnegative Perron eigenvector of $G$ has this property. Second, we answer a problem of Nikiforov (2009) by showing that the exact Tur\'an edge threshold is detected by the exact spectral threshold: for every $r\ge 2$ and every $n$, $\lambda(G)<\lambda(T_r(n))$, implying $e(G)<e(T_r(n)).$ Our proof also determines the equality cases. Third, we answer another question of Nikiforov (2009) by showing that his least-eigenvalue clique bound \[ \omega(G)\ge 1+\frac{2e(G)}{(n-d(G))(d(G)-\lambda_n(G))} \] does imply the concise form of Tur\'an's theorem. Finally, we discuss an open problem proposed by Ai et al. (2026) in \cite{ALNS26+}.

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arXiv2026

An improved double-exponential lower bound for $r_4(5,n)$

The Ramsey number $r_k(s,n)$ is the smallest integer $N$ such that every $N$-vertex $k$-graph contains either a copy of $K_s^{(k)}$ or an independent set of size $n$. A well-known conjecture of Erd\H{o}s and Hajnal states that for any fixed $4\le k<s$, $r_k(s,n)\ge \operatorname{twr}_{k-1}(\Omega(n)).$ At present, only the last two cases of this conjecture remain open, namely $r_4(5,n)\ge2^{2^{\Omega(n)}}$ and $r_4(6,n)\ge2^{2^{\Omega(n)}}$. Recently, Du, Hu, Liu, and Wang achieved a breakthrough by proving $r_4(5,n)\ge 2^{2^{\Omega(n^{1/7})}}$, which is the first double-exponential lower bound for $r_4(5,n)$. In this note, we improve this to $2^{2^{\Omega(n^{1/5})}}$ by modifying their construction and reducing the greedy selection of local maxima from seven layers to five, thereby making further progress towards the Erd\H{o}s-Hajnal conjecture.

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arXiv2026

An exponentially small gap of the Perron vector on independent sets

A classical result of Cioab\u{a} states that if $G$ is a connected graph with the unit Perron vector $\mathbf{x}$, then any independent set $S$ of $G$ satisfies $\sum_{v\in S} x_v^2 \le \frac{1}{2}$, with equality if and only if $G$ is a bipartite graph and $S$ is one of the partite sets. Let $\chi(G)= k $ be the chromatic number of $G$. A well-known conjecture of Gregory asserts that any independent set $S$ of $G$ satisfies $\frac{1}{2} - \sum_{v\in S}x_v^2 = \Omega ((k/n)^{1/2})$. Recently, Liu and Ning [J. Combin. Theory Ser. B 176 (2026)] disproved Gregory's conjecture by constructing a graph $G$ and an independent set $S$ such that $\frac{1}{2}- \sum_{v\in S}x_v^2 = O(k^5/n^3)$. Furthermore, they conjectured that this bound is tight up to a constant factor. In this paper, we first show that any cycle $C_n$ with odd integer $n\ge 7$ provides a simple counterexample to Gregory's conjecture. Second, we establish that for any independent set $S$, we have $\frac{1}{2} - \sum_{v\in S}x_v^2 = \frac{q}{4\lambda -2q}$, where $\lambda$ is the spectral radius of $G$, and $q$ is the Rayleigh quotient of $\mathbf{x}$ restricted to $\bar{S} :=V(G)\setminus S$. Third, we construct a graph with arbitrarily large chromatic number and find an independent set $S$ such that $\sum_{v\in S}x_v^2$ can be arbitrarily close to $\frac{1}{2}$, with an exponentially small gap. Our construction shows that there is no universal lower bound of the form $\Omega (k^{\alpha}/n^{\beta})$ for any $\alpha, \beta >0$. This settles both Gregory's original conjecture and the modified conjecture of Liu and Ning in the negative. Finally, we show the tightness of our construction and provide some local weighted lower bounds.

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arXiv2026

On Scott's odd induced subgraph conjecture and a related problem

For a graph $G$, let $f_o(G)$ denote the maximum order of an induced subgraph of $G$ all of whose vertices have odd degree, and let $\chi(G)$ denote the chromatic number of $G$. Scott (CPC, 1992) proved that $f_o(G) \ge |V(G)|/(2\chi(G))$ for every graph without isolated vertices, and conjectured that the factor $2$ can be removed. Wang and Wu (JGT, 2024) showed that this conjecture fails for bipartite graphs, but holds for line graphs. In this article, we confirm Scott's conjecture for claw-free graphs without isolated vertices, thereby strengthening the result of Wang and Wu. We also construct $K_{1,r}$-free graphs of arbitrarily large order to show that the conjecture fails for this broader class, for every integer $r \ge 4$. Wang and Wu also asked whether $f_o(L(G)) \ge n/2$ holds for every connected regular graph $G$ of order $n \ge 3$. We show that $C_5$ is the smallest counterexample to this problem. On the positive side, we prove that if $G$ is a connected $k$-regular $C_5$-free graph on $n$ vertices with $k \ge 2$, then $f_o(L(G)) \ge n/2$.

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arXiv2026

On derivatives and higher-order derivatives of chromatic polynomials

Let $ G $ be a graph of order $ n $ with maximum degree $\Delta$, and let $P(G,x)$ denote its chromatic polynomial. We investigate several properties of $P(G,x)$ related to its derivatives and higher-order derivatives. First, we study the monotonicity of $P(G,x)/x^n$. Dong proved that $(x-1)^nP(G,x)\geq x^nP(G,x-1)$ for all real $x\geq n$. In particular, taking $x=n$ establishes the Bartels-Welsh ``shameful conjecture" that $P(G,n)/P(G,n-1)>e$. Fadnavis later showed that the same inequality holds for all real $x\geq 36\Delta^{3/2}$. We improve this bound by proving that it also holds for all real $x\geq 10\Delta^{3/2}$. We then consider a conjecture of Dong, Ge, Gong, Ning, Ouyang, and Tay asserting that $ \frac{d^k}{dx^k} \bigl( \ln[(-1)^n P(G, x)] \bigr) < 0 $ for all $ k \geq 2 $ and $ x \in (-\infty, 0) $. We establish this conjecture for all $ k \geq 2 $ and $ x\leq -3.01\Delta k $.

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arXiv2026

On the chromatic profile for tripartite graphs and beyond

Let $H$ be a graph and let $\delta_{\chi}(H,r)$ denote the infimum of $c$ such that every $H$-free graph with minimum degree at least $cn$ is $r$-colorable. The \textit{chromatic profile} of $H$ is defined to be the values of $\delta_{\chi}(H,r)$ as $r$ varies. Erd\H{o}s and Simonovits described this graph parameter as ``too complicated", and Allen, B\"ottcher, Griffiths, Kohayakawa, and Morris posed its determination for every graph $H$ as an open problem \cite[Problem~45]{ABGKM2013}, emphasizing its expected difficulty. In this paper, we resolve the case $r=2$ for every graph $H$ with $\chi(H)=3$. We show that the set of possible values of $\delta_{\chi}(H,2)$ with $\chi(H)=3$ is finite and discrete: $$\{\delta_{\chi}(H,2):\chi(H)=3\}=\left\{\frac{1}{2},\frac{2}{5},\frac{2}{7},\frac{1}{4},\frac{2}{9},\frac{1}{5},\frac{2}{11},\frac{1}{6}\right\}.$$ Furthermore, we provide a complete structural characterization of the graphs $H$ associated with each threshold value. Moreover, we extend the classical chromatic profile result for triangle to color-critical graphs $H$ with $g_{\mathrm{odd}}(H)=\chi(H)=3$. Our approach introduces a useful auxiliary parameter. Motivated by the notion of vertex-extendability of Liu, Mubayi, and Reiher \cite{liu2023unified}, we define the {\it vertex-extendable threshold} of $H$, denoted by $\delta_{\mathrm{ext}}(H,r)$, as the infimum of $c\in (0,1)$ so that for every $H$-free graph $G$ on $n$ vertices, the existence of a vertex $v \in V(G)$ with $\chi(G - v) \leq r$ combined with $\delta(G)\ge cn$ implies that $G$ is $r$-colorable. A key structural consequence is that $\delta_{\chi}(H,2) = \max\left\{\delta_\chi(C_{2k+1},2),\delta_{\mathrm{ext}}(H,2)\right\},$ where $H$ is a color-critical graph with $\chi(H)=3$ and $g_{\mathrm{odd}}(H)=2k+1$ for $k\geq 2$.

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arXiv2026

Extensions of Erd\H{o}s's 1962 theorem on non-Hamiltonian graphs

For a positive integer $k$, a graph property $\mathcal{H}$, and a graph parameter $\mathcal{P}$, let $\operatorname{ex}_{\mathcal{P}}(n, \mathcal{H}; \delta \geq k)$ 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 $\operatorname{EX}_{\mathcal{P}}(n, \mathcal{H}; \delta \geq k)$. 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, Erd\H{o}s 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 Erd\H{o}s's 1962 theorem on non-Hamiltonian graphs. The first result gives a common generalization of the extremal theorem due to Erd\H{o}s 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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