Related papers: Large complete minors in expanding graphs
Lower bounds on the proof-theoretic strength of the graph minor theorem were found over 30 years ago by Friedman, Robertson and Seymour 1987, but upper bounds have always been elusive. We present recently found upper bounds on the graph…
A totally silver coloring of a graph G is a k--coloring of G such that for every vertex v \in V(G), each color appears exactly once on N[v], the closed neighborhood of v. A totally silver graph is a graph which admits a totally silver…
A vertex of a graph is bisimplicial if the set of its neighbors is the union of two cliques; a graph is quasi-line if every vertex is bisimplicial. A recent result of Chudnovsky and Seymour asserts that every non-empty even-hole-free graph…
A vertex subset $S$ of a graph $G$ is a perfect (resp. quasiperfect) dominating set in $G$ if each vertex $v$ of $G\setminus S$ is adjacent to only one vertex ($d_v\in\{1,2\}$ vertices) of $S$. Perfect and quasiperfect dominating sets in…
We give an improvement of a result of Zverovich and Zverovich which gives a condition on the first and last elements in a decreasing sequence of positive integers for the sequence to be graphic, that is, the degree sequence of a finite…
A graph is strongly perfect if every induced subgraph H has a stable set that meets every nonempty maximal clique of H. The characterization of strongly perfect graphs by a set of forbidden induced subgraphs is not known. Here we provide…
The existence problem of the total domination vertex critical graphs has been studied in a series of articles. The aim of the present article is twofold. First, we settle the existence problem with respect to the parities of the total…
Erd\H{o}s conjectured that every triangle-free graph $G$ on $n$ vertices contains a set of $\lfloor n/2 \rfloor$ vertices that spans at most $n^2 /50$ edges. Krivelevich proved the conjecture for graphs with minimum degree at least…
Gy\'arfas proved that every coloring of the edges of $K_n$ with $t+1$ colors contains a monochromatic connected component of size at least $n/t$. Later, Gy\'arf\'as and S\'ark\"ozy asked for which values of $\gamma=\gamma(t)$ does the…
We study convex entire graphs evolving with normal velocity equal to a positive power of the mean curvature. Under mild assumptions we prove longtime existence.
Kahale proved that linear sized sets in $d$-regular Ramanujan graphs have vertex expansion $\sim\frac{d}{2}$ and complemented this with construction of near-Ramanujan graphs with vertex expansion no better than $\frac{d}{2}$. However, the…
Large graphs are sometimes studied through their degree sequences (power law or regular graphs). We study graphs that are uniformly chosen with a given degree sequence. Under mild conditions, it is shown that sequences of such graphs have…
The main result of this paper is a proof of the continuity of a family of integral functionals defined on the space of functions of bounded variation with respect to a topology under which smooth functions are dense. These functionals occur…
We study extensions of direct sums of symmetric operators $S=\oplus_{n\in\mathbb{N}} S_n$. In general there is no natural boundary triplet for $S^*$ even if there is one for every $S_n^*$, $n\in\mathbb{N}$. We consider a subclass of…
We study sharp $p$-variational inequalities for the Hardy-Littlewood maximal operator on complete graphs, answering in the affirmative a question by Feng Liu and Qingying Xue. We also use computational assistance to find sharp constants in…
Using expander graphs, we construct a sequence of smooth compact surfaces with boundary of perimeter N, and with the first non-zero Steklov eigenvalue uniformly bounded away from zero. This answers a question which was raised in [9]. The…
We show that there is a constant c>0 so that for any fixed r which is at least 3 a.a.s. an r-regular graph on n vertices contains a complete graph on c n^{1/2} vertices as a minor. This confirms a conjecture of Markstrom. Since any minor of…
We study the evolution of convex complete non-compact graphs by positive powers of Gauss curvature. We show that if the initial complete graph has a local uniform convexity, then the graph evolves by any positive power of Gauss curvature…
Let $G$ be a graph, and $H\colon V(G)\to 2^\mathbb{N}$ a set function associated with $G$. A spanning subgraph $F$ of $G$ is called an $H$-factor if the degree of any vertex $v$ in $F$ belongs to the set $H(v)$. This paper contains two…
Given a constant $\alpha>0$, an $n$-vertex graph is called an $\alpha$-expander if every set $X$ of at most $n/2$ vertices in $G$ has an external neighborhood of size at least $\alpha|X|$. Addressing a question posed by Friedman and…