Related papers: Cycle lengths in expanding graphs

Given a graph $G$, the strong clique number $\omega_2'(G)$ of $G$ is the cardinality of a largest collection of edges every pair of which are incident or connected by an edge in $G$. We study the strong clique number of graphs missing some…

A minimum feedback arc set of a directed graph $G$ is a smallest set of arcs whose removal makes $G$ acyclic. Its cardinality is denoted by $\beta(G)$. We show that an Eulerian digraph with $n$ vertices and $m$ arcs has $\beta(G) \ge…

A graph $G$ has a $C_k$-decomposition if its edge set can be partitioned into cycles of length $k$. We show that if $\delta(G)\geq 2|G|/3-1$, then $G$ has a $C_4$-decomposition, and if $\delta(G)\geq |G|/2$, then $G$ has a…

We show that every graph $G$ on $n$ vertices with $\delta(G) \geq (1/2+\varepsilon)n$ is spanned by a complete blow-up of a cycle with clusters of nearly uniform size $\Omega(\log n)$. The proof is based on a recently introduced approach…

This paper studies thresholds in random generalized Johnson graphs for containing large cycles, i.e. cycles of variable length growing with the size of the graph. Thresholds are obtained for different growth rates.

Let $\Gamma(n,k)$ be the set of $2$-connected $n$-vertex graphs containing an edge that is not on any cycle of length at least $k+1.$ Let $g_s(n,k)$ denote the maximum number of $s$-cliques in a graph in $\Gamma(n,k).$ Recently, Ji and Ye…

A straight-line drawing $\delta$ of a planar graph $G$ need not be plane, but can be made so by \emph{untangling} it, that is, by moving some of the vertices of $G$. Let shift$(G,\delta)$ denote the minimum number of vertices that need to…

Call a simple graph $H$ of order $n$ well-separable, if by deleting a separator set of size $o(n)$ the leftover will have components of size at most $o(n)$. We prove, that bounded degree well-separable spanning subgraphs are easy to embed:…

We present improved algorithms for short cycle decomposition of a graph. Short cycle decompositions were introduced in the recent work of Chu et al, and were used to make progress on several questions in graph sparsification. For all…

For a graph $G$ with adjacency matrix $A(G)$ and degree diagonal matrix $D(G)$, the $A_{\alpha}$-matrix of $G$ is defined as \begin{equation*} A_{\alpha}(G) = \alpha D(G) + (1- \alpha) A(G), \text{ for any } \alpha \in [0,1].…

A graph G is (a:b)-colorable if there exists an assignment of b-element subsets of {1,...,a} to vertices of G such that sets assigned to adjacent vertices are disjoint. We show that every planar graph without cycles of length 4 or 5 is…

A graph $G=(V,E)$ is said to be a \textit{$k$-threshold graph} with \textit{thresholds} $\theta_1<\theta_2<...<\theta_k$ if there is a map $r: V \longrightarrow \mathbb{R}$ such that $uv\in E$ if and only if $\theta_i\le r(u)+r(v)$ holds…

Given integers $n\ge \Delta\ge 2$, let $\mathcal{T}(n, \Delta)$ be the collection of all $n$-vertex trees with maximum degree at most $\Delta$. A question of Alon, Krivelevich and Sudakov in 2007 asks for determining the best possible…

In 1975, P. Erd\H{o}s proposed the problem of determining the maximum number $f(n)$ of edges in a graph on $n$ vertices in which any two cycles are of different lengths. Let $f^{\ast}(n)$ be the maximum number of edges in a simple graph on…

We introduce a new technique for bounding the cover time of random walks by relating it to the runtime of randomized broadcast. In particular, we strongly confirm for dense graphs the intuition of Chandra et al. \cite{CRRST97} that "the…

Let $G$ be a graph. A good function is a function $f:V(G)\rightarrow \{-1,1\}$, satisfying $f(N(v))\geq 1$, for each $v\in V(G)$, where $ N(v)=\{u\in V(G)\, |\, uv\in E(G) \} $ and $f(S) = \sum_{u\in S} f(u)$ for every $S \subseteq V(G) $.…

It is proved that if $G$ is a $t$-tough graph of order $n$ and minimum degree $\delta$ with $t>1$ then either $G$ has a cycle of length at least $\min\{n,2\delta+4\}$ or $G$ is the Petersen graph.

An identifying open code of a graph $G$ is a set $S$ of vertices that is both a separating open code (that is, $N_G(u) \cap S \ne N_G(v) \cap S$ for all distinct vertices $u$ and $v$ in $G$) and a total dominating set (that is, $N(v) \cap S…

For a connected $n$-vertex graph $G$ and a set $\mathcal{F}$ of graphs, let $\iota(G,\mathcal{F})$ denote the size of a smallest set $D$ of vertices of $G$ such that the graph obtained from $G$ by deleting the closed neighbourhood of $D$…

Let ${\rm dim}(G)$ and $D(G)$ respectively denote the metric dimension and the distinguishing number of a graph $G$. It is proved that $D(G) \le {\rm dim}(G)+1$ holds for every connected graph $G$. Among trees, exactly paths and stars…