Related papers: A lower bound on the average degree forcing a mino…

A graph $G$ is \textit{$k$-critical} if $\chi(G) = k$ and every proper subgraph of $G$ is $(k - 1)$-colorable, and if $L$ is a list-assignment for $G$, then $G$ is \textit{$L$-critical} if $G$ is not $L$-colorable but every proper induced…

The "slope-number" of a graph $G$ is the minimum number of distinct edge slopes in a straight-line drawing of $G$ in the plane. We prove that for $\Delta\geq5$ and all large $n$, there is a $\Delta$-regular $n$-vertex graph with…

Mader proved that every sufficiently large graph with average degree at least $(2+\sqrt{2})k$ has a $(k+1)$-connected subgraph. He also conjectured that an average degree of at least $3k$ is sufficient. The best known sufficient factor was…

Let $\lambda(G)$ be the smallest number of vertices that can be removed from a non-empty graph $G$ so that the resulting graph has a smaller maximum degree. Let $\lambda_{\rm e}(G)$ be the smallest number of edges that can be removed from…

We study minimum degree conditions that guarantee that an $n$-vertex graph is rigid in $\mathbb{R}^d$. For small values of $d$, we obtain a tight bound: for $d = O(\sqrt{n})$, every $n$-vertex graph with minimum degree at least $(n+d)/2 -…

Let $G$ be a graph on $n$ vertices and $(H,+)$ be an abelian group. What is the minimum size ${\sf S}_H(G)$ of the set of all sums $A(u)+A(v)$ over all injections $A:V(G)\to H$? In 2012, the first author, Angel, the second author, and…

Mubayi and Verstraete conjectured that if $T$ is a tree on $t + 1$ vertices, then any $n$-vertex graph $G$ with average degree $d$ contains at least \[ n d(d - 1) \cdots (d - t + 1) \] labeled copies of $T$ as long as $d$ is sufficiently…

Let $G$ be a finite, simple, and undirected graph of order $n$ and average degree $d$. Up to terms of smaller order, we characterize the minimal intervals $I$ containing $d$ that are guaranteed to contain some vertex degree. In particular,…

We analyse an extremal question on the degrees of the link graphs of a finite regular graph, that is, the subgraphs induced by non-trivial spheres. We show that if $G$ is $d$-regular and connected but not complete then some link graph of…

A graph $ G $ is minimally $ t $-tough if the toughness of $ G $ is $ t $ and deletion of any edge from $ G $ decreases its toughness. Katona et al. conjectured that the minimum degree of any minimally $ t $-tough graph is $ \lceil 2t\rceil…

Let C(G) denote the set of lengths of cycles in a graph G. In the first part of this paper, we study the minimum possible value of |C(G)| over all graphs G of average degree d and girth g. Erdos conjectured that |C(G)| =\Omega(d^{\lfloor…

In 1972 Mader proved that every graph with average degree at least $4k$ has a $(k+1)$-connected subgraph with more than $2k$ vertices. We improve this bound by showing that the constant $4$ can be replaced by $3+\frac{1}{3}$; this bound is…

For any $\alpha\in (0,1)$ and any $n^{\alpha}\leq d\leq n/2$, we show that $\lambda(G)\leq C_\alpha \sqrt{d}$ with probability at least $1-\frac{1}{n}$, where $G$ is the uniform random $d$-regular graph on $n$ vertices, $\lambda(G)$ denotes…

We prove that every $n$-vertex directed graph $G$ with the minimum outdegree $\delta^+(G) = d$ contains a subgraph $H$ satisfying \[ \min\left\{\delta^+(H), \delta^-(H) \right\} \ge \frac{d(d+1)}{2n} \,.\] We also show that if $d = o(n)$…

We prove the following Alon-Boppana type theorem for general (not necessarily regular) weighted graphs: if $G$ is an $n$-node weighted undirected graph of average combinatorial degree $d$ (that is, $G$ has $dn/2$ edges) and girth $g>…

In this note we give a new upper bound for the Laplacian eigenvalues of an unweighted graph. Let $G$ be a simple graph on $n$ vertices. Let $d_{m}(G)$ and $\lambda_{m+1}(G)$ be the $m$-th smallest degree of $G$ and the $m+1$-th smallest…

The Randi{\' c} index of a graph $G$, written $R(G)$, is the sum of $\frac 1{\sqrt{d(u)d(v)}}$ over all edges $uv$ in $E(G)$. %let $R(G)=\sum_{uv \in E(G)} \frac 1{\sqrt{d(u)d(v)}}$, which is called the Randi{\' c} index of it. Let $d$ and…

A set of vertices in a graph is a Hamiltonian subset if it induces a subgraph containing a Hamiltonian cycle. Kim, Liu, Sharifzadeh and Staden proved that among all graphs with minimum degree $d$, $K_{d+1}$ minimises the number of…

We show that every graph $G$ of maximum degree $\Delta$ and sufficiently large order has a vertex cutset $S$ of order at most $\Delta$ that induces a subgraph $G[S]$ of maximum degree at most $\Delta-3$. For $\Delta\in \{ 4,5\}$, we refine…

Estimating the average degree of graph is a classic problem in sublinear graph algorithm. Eden, Ron, and Seshadhri (ICALP 2017, SIDMA 2019) gave a simple algorithm for this problem whose running time depended on the graph arboricity, but…