A family of paths in a graph $G$ strongly separates the edges of $G$ if for every ordered pair of distinct edges $(e,f)$ some path in the family contains $e$ and avoids $f$; the minimum size of such a family is denoted by $\operatorname{ssp}(G)$. Bonamy, Botler, Dross, Naia, and Skokan proved, for every $n$-vertex graph $G$, that $\operatorname{ssp}(G)\le 19n$; Liu, Xu, and Yang recently improved this to $\operatorname{ssp}(G)\le 10n-o(n)$. We prove, for every $n$-vertex graph $G$, that $\operatorname{ssp}(G)\le 7.75n$.
An $n$-vertex graph $G$ is $(\rho, d)$-dense if for all $U \subseteq V(G)$ of size at least $\rho n$, the induced subgraph $G[U]$ contains at least $\frac{d}{2}|U|^2$ edges. In 2010, partly motivated by the theory of quasirandomness, Kohayakawa, Nagle, R\"{o}dl, and Schacht conjectured that, for any fixed graph $F$, th...
A set $D\subseteq V(G)$ is a $1$-limited dominating set of a graph $G$ if every vertex outside $D$ has a neighbor in $D$ and every vertex of $D$ has at most one neighbor outside $D$. The minimum cardinality among all $1$-limited dominating sets of $G$ is denoted by $\gamma_1^{\mathrm L}(G)$. For every cubic graph $G$ o...
Goran Radić, Gordana Radić, Aleksandra Tepeh· 0 citations
For a graph $G$, let $e(G)$ denote the number of edges of $G$, and let $c(G)$ be the number of distinct cycles in $G$. Morrison, Roberts and Scott asked whether, for every fixed graph $H$ and all large $n$, some $n$-vertex $H$-free graph maximizes both $e(G)$ and $c(G)$. In this paper, we show that the answer is no for...
A path in a properly edge-colored graph is rainbow if its edges have pairwise distinct colors. For a proper edge-coloring $c$ of a graph $G$, let $\operatorname{rpc}(G,c)$ be the minimum number of rainbow paths needed to cover $E(G)$, and let $\operatorname{rpc}(G)$ be the maximum of $\operatorname{rpc}(G,c)$ over all...
Let $\mathbf{G}=\{G_1,\dots,G_{n-1}\}$ be a collection of not necessarily distinct graphs of order $n$ on the same vertex set $V$. A path $P$ with $V(P)\subseteq V$ is called \emph{rainbow} in $\mathbf{G}$, if there exists an injection $\phi\colon E(P)\to [n-1]$ such that $e\in E(G_{\phi(e)})$ for each $e\in E(P)$. In...
For a graph $G$, let $f_t(G)$ denote the minimum of the maximum degree of an induced subgraph with $\alpha(G)+t$ vertices, where $\alpha(G)$ is the independence number, and write $f(G)=f_1(G)$. Huang's theorem gives $f(Q_k)\ge\lceil\sqrt{k}\rceil$ for the $k$-dimensional hypercube $Q_k$. We extend this lower bound to C...
