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.
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...
For ordered graphs $H_1,\ldots,H_t$, let $\rt(H_1,\ldots,H_t)$ denote the least integer $N$ such that every $t$-coloring of the edges of the naturally ordered complete graph on $[N]$ contains an ordered copy of $H_i$ in color $i$ for some $i\in[t]$. We prove that a uniformly random ordered matching $M$ on $n$ vertices...
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...
For a graph $G$, let $\tau_{\mathrm{iso}}(G)$ denote the number of isomorphism classes of its spanning trees. For every fixed $d\ge3$ and all sufficiently large $n$, we prove that every connected $n$-vertex graph $G$ with $\delta(G)\ge d$ satisfies \[\tau_{\mathrm{iso}}(G)\ge \tau_{\mathrm{iso}}(K_{d,n-d})=A_dn^{d-1}+O...
Let $G$ be a graph together with a total order $\prec$ on its edges. We say that $\prec$ is realizable in $\mathbb{R}^d$ if there is a placement of the vertices of $G$ in $\mathbb{R}^d$ such that the Euclidean lengths of the edges induce exactly the order $\prec$. Almendra-Hern\'andez and Mart\'inez-Sandoval proved tha...
Gerardo L. Maldonado, Leonardo Martínez-Sandoval, Miguel Raggi et al.· 0 citations
Let $r\geq3$ be fixed, and let $\mathbf{G}_n$ be the set of all simple graphs with vertex set $[n]=\{1,\ldots,n\}$. We consider an exponential random graph model which gives higher probability to $G \in \mathbf{G}_n$ than to $H \in \mathbf{G}_n$ if $G$ has fewer $r$-cliques than $H$. But all graphs in $\mathbf{G}_n$ ha...
Yasmin Tousinejad, Vera Koponen· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
