Related papers: Maximum degree in minor-closed classes of graphs

A random geometric digraph $G_n$ is constructed by taking $\{X_1,X_2,... X_n\}$ in $\mathbb{R}^2$ independently at random with a common bounded density function. Each vertex $X_i$ is assigned at random a sector $S_i$ of central angle…

This work focuses on the degree bound of maps between balls with maximum geometric rank and minimum target dimension where this geometric rank occurs. Specifically, we show that rational proper maps between $\mathbb{B}_n$ and $\mathbb{B}_N$…

Let $\mathcal G$ be a separable family of graphs. Then for all positive constants $\epsilon$ and $\Delta$ and for every sufficiently large integer $n$, every sequence $G_1,\dotsc,G_t\in\mathcal G$ of graphs of order $n$ and maximum degree…

Consider a graph $G=(V,E)$ without isolated edges and with maximum degree $\Delta$. Given a colouring $c:E\to\{1,2,\ldots,k\}$, the weighted degree of a vertex $v\in V$ is the sum of its incident colours, i.e., $\sum_{e\ni v}c(e)$. For any…

Let $G$ be a connected graph of order $n$.The Wiener index $W(G)$ of $G$ is the sum of the distances between all unordered pairs of vertices of $G$. In this paper we show that the well-known upper bound $\big( \frac{n}{\delta+1}+2\big) {n…

The domatic number of a graph $G$, denoted $dom(G)$, is the maximum possible cardinality of a family of disjoint sets of vertices of $G$, each set being a dominating set of $G$. It is well known that every graph without isolated vertices…

We prove that a class of graphs with an excluded minor and with the maximum degree sublinear in the number of edges is maximally modular, that is, modularity tends to 1 as the number of edges tends to infinity.

A graph $G$ is terminal-pairable with respect to a demand multigraph $D$ on the same vertex set as $G$, if there exists edge-disjoint paths joining the end vertices of every demand edge of $D$. In this short note, we improve the upper bound…

Let $G$ be a connected uniform hypergraphs with maximum degree $\Delta$, spectral radius $\lambda$ and minimum H-eigenvalue $\mu$. In this paper, we give some lower bounds for $\Delta-\lambda$, which extend the result of [S.M. Cioab\u{a},…

For a fixed degree sequence $\mathcal{D}=(d_1,...,d_n)$, let $G(\mathcal{D})$ be a uniformly chosen (simple) graph on $\{1,...,n\}$ where the vertex $i$ has degree $d_i$. In this paper we determine whether $G(\mathcal{D})$ has a giant…

We show that for any fixed dense graph G and bounded-degree tree T on the same number of vertices, a modest random perturbation of G will typically contain a copy of T . This combines the viewpoints of the well-studied problems of embedding…

A graph $G$ is said to have \textit{bandwidth} at most $b$, if there exists a labeling of the vertices by $1,2,..., n$, so that $|i - j| \leq b$ whenever $\{i,j\}$ is an edge of $G$. Recently, B\"{o}ttcher, Schacht, and Taraz verified a…

This work will appear as a chapter in a forthcoming volume titled "Topics in Probabilistic Graph Theory". A theory of scaling limits for random graphs has been developed in recent years. This theory gives access to the large-scale geometric…

Let $G$ be a $t$-tough graph of order $n$ and minimum degree $\delta$ with $t>1$. It is proved that if $\delta\ge(n-2)/3$ then each longest cycle in $G$ is a dominating cycle.

Developing further Stein's recent notion of relative end degrees in infinite graphs, we investigate which degree assumptions can force a locally finite graph to contain a given finite minor, or a finite subgraph of given minimum degree.…

The maximum likelihood threshold (MLT) of a graph $G$ is the minimum number of samples to almost surely guarantee existence of the maximum likelihood estimate in the corresponding Gaussian graphical model. We give a new characterization of…

Let G be a family of n-vertex graphs of uniform degree 2 with the property that the union of any two member graphs has degree four. We determine the leading term in the asymptotics of the largest cardinality of such a family. Several…

The minimum positive $\ell$-degree $\delta^+_{\ell}(G)$ of a non-empty $k$-graph $G$ is the maximum $m$ such that every $\ell$-subset of $V(G)$ is contained in either none or at least $m$ edges of $G$; let $\delta^+_{\ell}(G):=0$ if $G$ has…

The maximum likelihood threshold (MLT) of a graph $G$ is the minimum number of samples to almost surely guarantee existence of the maximum likelihood estimate in the corresponding Gaussian graphical model. Recently a new characterization of…

Given a graph $F$, the random Tur\'an problem asks to determine the maximum number of edges in an $F$-free subgraph of $G_{n,p}$. Prior to this work, the only bipartite graphs $F$ with known tight bounds included certain classes of complete…