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
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 $\mathsc...
The power graph $\mathscr{P}(G)$ of a group $G$ is defined as the simple graph with vertex set $G$, and two distinct vertices $x$ and $y$ are joined by an edge if and only if either $x= y^k$ or $y= x^k$, $k \in \mathbb{N}$. In this paper, first, we obtain the characteristic polynomial of $\mathscr{P}(\mathbb{Z}_m \time...
Komal Kumari, P. Panigrahi· Journal of Algebra Combinato...· 0 citations
For a family $\mathcal{F}$ of $k$-graphs, $\ex_k(n,\mathcal{F})$ denotes the maximum number of edges in an $n$-vertex $\mathcal{F}$-free $k$-graph. Let $M_{s+1}^k$ denote a matching of size $s+1$ in $k$-uniform hypergraphs. Recently, Alon and Frankl (JCTB, 2024) determined $\ex_2(n,\{M_{s+1}^2,K_{\ell+1}\})$ for all $n...
Let $(G_n)_{n\ge n_0}$ be a family of graphs on $[n]$ whose edge ideals $I_n\subseteq R_n=k[x_1,\dots,x_n]$ form an $\mathrm{Inc}(\mathbb{N})$-invariant chain, $I_{n+r}=\mathrm{Inc}(\mathbb{N})_{n,n+r}(I_n)$ for $n\ge n_0,\ r\ge0$. We determine which pairs $(n,r)$ make $R_{n+r}/I_{n+r}$ Cohen--Macaulay, for five classi...
Imran Anwar, Mughees Ghayas, A. Javed· 0 citations
The $n$-dimensional hypercube $Q_n$ is the graph with vertex set $\{0,1\}^n$ in which two vertices are adjacent if they differ in exactly one coordinate. For a graph $H$, let $\operatorname{ex}(Q_n,H)$ be the maximum number of edges in an $H$-free subgraph of $Q_n$. The hypercube Tur\'an density of $H$ is defined by $\...
Let $M=(E,\mathcal{I})$ be a matroid of rank $r$. Let $\mathcal{I}_k$ be the independent sets of size $k$, and let $I_k=|\mathcal{I}_k|$ and $I=|\mathcal{I}|$. We show that if every set $F\in \mathcal{I}_{r-1}$ is contained in at least $\delta$ bases, then $$\ln \left(\frac{I}{I_r}\right)\geqslant \frac{I_{r-1}}{I_r}\c...
Ferenc Bencs, P. Csikvári· 0 citations
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