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.
For an $n$-vertex graph $G$ and a permutation $\sigma$ of its vertex set, let $\sigma(G)$ denote the corresponding relabelling of $G$, and put \[ I_G(\sigma)=|E(G)\cap E(\sigma(G))|. \] Let $f(n,k)$ be the minimum number of edges in an $n$-vertex graph for which $I_G(\sigma)\geq k$ for every $\sigma$. In 1977 Erd\H{o}s...
For a graph $G$, let $L(G)=\max\{\omega(G),\lceil (|V(G)|+1)/(\alpha_{\min}(G)+1)\rceil\}$, where $\omega(G)$ is the clique number and $\alpha_{\min}(G)$ is the minimum, over all vertices $v$, of the largest size of an independent set containing $v$. Dybizba\'nski, Furma\'nczyk, and Mkrtchyan (Discrete Appl. Math. 354...
Juho Lauri· Discrete Applied Mathematics· 0 citations
For an $n$-vertex graph $G$ and a permutation $\pi$ of its vertex set, let \[ I_G(\pi)=|E(G)\cap E(G_{\pi})|,\qquad \mu(G)=\min_{\pi} I_G(\pi), \] where $G_{\pi}$ is the copy of $G$ obtained by relabelling every vertex $x\in V(G)$ as $\pi(x)$. Let $f(n,k)$ be the minimum number of edges in an $n$-vertex graph $G$ satis...
For a connected and non-complete graph $G$ of order $n$, its toughness is defined as \[ \tau(G)=\min\bigl\{|S|/c(G-S):S\subseteq V(G),\ c(G-S)>1\bigr\}, \] where $c(G-S)$ denotes the number of components of $G-S$. Let $\alpha(G)$ denote the independence number of $G$. An elementary bound on toughness is $$\tau(G)\leq\f...
For an $F$-free graph $G$, a non-edge is $F$-saturating if adding it to $G$ creates a copy of $F$. We denote by $f_{p+1}(n,m)$ the minimum number of $K_{p+1}$-saturating non-edges in a $K_{p+1}$-free $n$-vertex graph with $m$ edges. Erd\H{o}s and Tuza conjectured that $f_4\left(n,\mathrm{ex}(n,K_3)+ 1\right)= (1 + o(1)...
In this paper we introduce and explore the following new graph invariant: For a graph $G$ on $k$ vertices, $G \neq K_k$, let $st(n,G)$ denote the maximum number of edges in a graph of order $n$ which does not contain any subgraph on $k$ vertices strictly containing $G$. A basic relation to classical Tur\'an numbers is...
Y. Caro, Z. Tuza· 0 citations
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