For a graph $H$ and a family of graphs $\mathcal F$, let $\text{ex}(n,H,\mathcal F)$ denote the maximum number of copies of $H$ in an $\mathcal F$-free graph on $n$ vertices. For every integer $i\ge 3$, let $C_i$ denote the cycle of length $i$. For $r\ge 3$, set $\mathscr {C}_r=\{C_3,C_4,\ldots,C_r\},$ and set $\mathscr {C}_2=\varnothing$. In this paper, we prove that, for all integers $l>k\ge 2$, $$ \text{ex}(n,C_{2k+1},\mathscr {C}_{2k}\cup\{C_{2l+1}\}) =O_{k,l} \left(n^{2-\frac{1}{k(k+1)(l-k)}}\ \ \right). $$ Together with the known upper bounds for the number of triangles in $C_{2l+1}$-free graphs, this confirms a conjecture of Gerbner, Gy\H{o}ri, Methuku, and Vizer.
For graphs $G$ and $H$, let $\mathbf N(G,H)$ denote the number of unlabeled, not necessarily induced copies of $H$ in $G$, and let $\mathbf N_{\mathcal P}(n,H)$ be the maximum of $\mathbf N(G,H)$ over all $n$-vertex planar graphs $G$. We prove that, for every fixed integer $m\geq 3$, $$\mathbf N_{\mathcal P}(n,C_{2m+1})=2m\left(\frac{n}{m}\right)^m+O_m\!\left(n^{m-1/5}\right).$$ The proof uses a sharp weighted cycle--path inequality for edge probability measures on finite complete graphs. This strengthens a conjecture of Heath, Martin, and Wells and, together with their reduction lemma, yields the stated asymptotic formula.
Let $\mathcal{C}_{[4,r]}$ be the family of cycles $\{C_4, \dots, C_r\}$. A graph $G$ is said to be $\mathcal{C}_{[4,r]}$-saturated if $G$ does not contain a copy of cycle $C_i$ for $4\le i\le r$, but the addition of any edge $e\notin E(G)$ creates at least one copy of $C_i$ for $4\le i\le r.$ The saturation number $sat(n, \mathcal{C}_{[4,r]})$ is the minimum number of edges in an $n$-vertex $\mathcal{C}_{[4,r]}$-saturated graph. In 2025, Ma determined that $sat(n, \mathcal{C}_{[4,5]})=\lceil \frac{5n}{4} - \frac{3}{2} \rceil$, and conjectured that for any $r \ge 5$, $sat(n, \mathcal{C}_{[4,r]}) = \lceil\frac{5n}{4} - \frac{3}{2} \rceil$ holds for large $n$. In this paper we prove that $sat(n, \mathcal{C}_{[4,r]}) \le \lceil\frac{5n}{4} - \frac{r+1}{4}\rceil$ for $n \ge r+1$, which disproves Ma's conjecture for $r\ge 6.$ For $r=6,$ we determine that $sat(n, \mathcal{C}_{[4,6]})=\lceil\frac{5n}{4}-\frac{7}{4}\rceil.$ {\bf Keywords}: Saturation graphs; Saturation number; Cycles; Edge minimization
Given a graph $G$ with degree sequence $d_{1},\ldots,d_{n}$ and a positive real number $p$, let $e_{p}(G)=\sum_{i=1}^{n} d_{i}^{p}$. For a fixed family of graphs $\mathcal F$, let $ex_{p}(n, \mathcal F)$ denote the maximum value of $e_{p}(G)$ over all $\mathcal F$-free graphs $G$ on $n$ vertices. In 2000, Caro and Yuster introduced the following Tur\'an-type problem: For a positive integer $p$ and a fixed graph $F$, determine $ex_{p}(n, F)$, and characterize the extremal graphs $G$ on $n$ vertices that attain $ex_p(n, F)$. Recently, Gao, Liu, Ma and Pikhurko proved that $ex_{p}(n, \mathcal F)=(\tau(\mathcal F)-1+o(1))n^p$ for real $p>\frac{1}{1-\alpha}$, where $\mathcal F$ is a degenerate family of graphs with classical Tur\'an number $ex(n, \mathcal F)=O(n^{1+\alpha})$ for some $\alpha\in[0,1)$, and $\tau(\mathcal F)$ is the minimum size of an independent vertex cover over all bipartite graphs $F\in\mathcal F$. Based on their method, we obtain a stability result for $ex_{p}(n, \mathcal F)$, and prove that all extremal graphs must contain the complete bipartite graph $K_{\tau(\mathcal F)-1,n-\tau(\mathcal F)+1}$ when $n$ is sufficiently large. Our results can be used to deduce all previously known results about $ex_{p}(n, F)$ when $F$ is a bipartite graph and $n$ is sufficiently large. We also obtain several new exact results for $ex_{p}(n, F)$, namely, when $F$ is an even cycle, a complete bipartite graph, a discrete hypercube, a caterpillar forest, and a spider forest.
For a graph $H$ with $3\mid e(H)$, the zero-sum Ramsey number $R(H,\Z_3)$ is the least integer $N$ such that every labeling of the edges of $K_N$ by elements of $\Z_3$ contains a copy of $H$ whose edge labels sum to zero. We determine the last previously unresolved infinite family in the complete-graph case modulo $3$. More precisely, we prove that \(R(K_n,\Z_3)=n+3\) for every $n\ge 10$ satisfying $n\equiv 1\pmod 3$. Consequently, for $k\ge 1$, \(R(K_{9k+7},\Z_3)=9k+10\), resolving a problem of Caro and Mifsud.
Let $G$ be a simple graph on $n$ vertices and $e(G)$ edges. Let $\mu_1\geq \cdots \geq \mu_{n-1}\geq \mu_n=0$ be the Laplacian eigenvalues of $G$. For $k=1, \ldots, n$, let $S_k(G)=\sum_{i=1}^{k}\mu_i$. Brouwers conjecture asserts that for any $k\in\{1,\ldots, n\}$, $S_k(G)\leq e(G)+\binom{k+1}{2}$. In [Bounding the sum of the largest Laplacian eigenvalues of graphs, {\em Discrete Appl. Math.}, 170:95--103, (2014)], Rocha and Trevisan showed that the conjecture holds true for $1 \leq k \leq \lfloor g/5 \rfloor$, where $g$ denotes the girth of $G$. This bound on $k$ was later improved by Chen in [Improved results on Brouwers conjecture for sum of the Laplacian eigenvalues of a graph, {\em Linear Algebra Appl.}, 557:327--338, (2018)], who established that the conjecture holds for $1 \leq k \leq \lfloor g/4 \rfloor$. In this article, we further strengthen these results by proving that the Brouwers conjecture holds for $1\leq k\leq\left\lfloor \frac{m(G)}{2}\right\rfloor,$ where $m(G)$ denotes the matching number of $G$. Since $m(G)\geq \lfloor g/2\rfloor$, the case constitutes a genuine improvement over the aforementioned results. As an application, we show that if $G$ is a graph of order $n$ and with a perfect matching, then the Brouwers conjecture holds for $1\leq k\leq \left\lfloor \frac{n}{4}\right\rfloor$. Finally, we provide a new perspective on verifying Brouwers conjecture by proving that $G$ satisfies the Brouwers conjecture if and only if for a fixed positive integer $h$, $\mathcal{S}_h(\overline{G})\leq e(\overline{G})+\binom{h+1}{2}$ holds whenever $\mathcal{S}_h(G)\leq e(G)+\binom{h+1}{2}$, where $\overline{G}$ denotes the complement of $G$.
Given graphs $H$ and $F$, the generalized Tur\'{a}n number ${\rm ex}(n,H,F)$ is the maximum number of copies of $H$ in an $n$-vertex $F$-free graph. Alon and Shikhelman (J. Combin. Theory Ser. B, 2016) initiated the systematic study of generalized Tur\'{a}n problems. Recently, Gao, Wu and Xue (J. Graph Theory, 2026) asked whether every graph $F$ with chromatic number $\chi(F)=r\geq3$ and treewidth ${\rm tw}(F)\geq r$ satisfies ${\rm ex}(n,K_r,F)=\Omega(n^{r-1})$. In this note, we give a negative answer to this question for every $r\geq3$. More precisely, we prove that the graph $F_r=K_{r-3}\vee H$, where $H$ is obtained from $K_4$ by subdividing one edge once, satisfies $\chi(F_r)={\rm tw}(F_r)=r$ and \[ n^{r-1}e^{-O(\sqrt{\log n})}\leq {\rm ex}(n,K_r,F_r)=o(n^{r-1}). \] This result also disproves Conjecture 6.3 in the recent survey of Gerbner and Palmer (Electron. J. Combin., 2026).
Junpeng Zhou, Xi-Ying Yuan· 0 citations
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