Related papers: A graph counterexample to davies' conjecture

Reed conjectured that for every graph, $\chi \leq \left \lceil \frac{\Delta + \omega + 1}{2} \right \rceil$ holds, where $\chi$, $\omega$ and $\Delta$ denote the chromatic number, clique number and maximum degree of the graph, respectively.…

A \emph{proportionally dense subgraph} (PDS) is an induced subgraph of a graph with the property that each vertex in the PDS is adjacent to proportionally as many vertices in the subgraph as in the rest of the graph. In this paper, we study…

The aim of this note is to show that lamplighter graphs where the space graph is infinite and at most two-ended and the lamp graph is at most two-ended do not admit harmonic functions with gradients in $\ell^p$ (\ie finite $p$-energy) for…

We prove that the hairy graph complex with the extra differential is almost acyclic. This justifies the construction of hairy graph cohomology classes by the waterfall mechanism. The main idea used in the paper is a new differential,…

A meteor graph is a connected graph with no sources and sinks consisting of two disjoint cycles and the paths connecting these cycles. We prove that two meteor graphs are shift equivalent if and only if they are strongly shift equivalent,…

We construct infinitely many connected, circulant digraphs of outdegree three that have no hamiltonian circuit. All of our examples have an even number of vertices, and our examples are of two types: either every vertex in the digraph is…

We consider undirected graphs that arise as deterministic functions of stationary point processes such that each point has degree bounded by two. For a large class of point processes and edge-drawing rules, we show that the arising graph…

We consider the Dirichlet form given by \sE(f,f)&=&{1/2}\int_{\bR^d}\sum_{i,j=1}^d a_{ij}(x)\frac{\partial f(x)}{\partial x_i} \frac{\partial f(x)}{\partial x_j} dx &+&\int_{\bR^d\times \bR^d} (f(y)-f(x))^2J(x,y)dxdy. Under the assumption…

Given integers $\Delta\ge 2$ and $t\ge 2\Delta$, suppose there is a graph of maximum degree $\Delta$ and a partition of its vertices into blocks of size at least $t$. By a seminal result of Haxell, there must be some independent set of the…

The present note concerns the "graph of graphs" that has cubic graphs as vertices connected by edges represented by the so-called Whitehead moves. Here, we prove that the outer-conductance of the graph of graphs tends to zero as the number…

Let v(G) and dom(G) denote the number of vertices and the domination number of a graph G, and let r (G) = dom(G)/v(G)$. Let [x] and ]x[ be the floor and the ceiling of a number x. In 1996 B. Reed conjectured that if G is a cubic graph, then…

Let f be a function mapping an n dimensional vector space over GF(p) to GF(p). When p is 2, Bernasconi et al. have shown that there is a correspondence between certain properties of f (e.g., if it is bent) and properties of its associated…

A class of countable infinite graphs with unbounded vertex degree is considered. In these graphs, the vertices of large degree `repel' each other, which means that the path distance between two such vertices cannot be smaller than a certain…

A connected graph $G$ with chromatic number $t$ is double-critical if $G \backslash \{x, y\}$ is $(t - 2)$-colorable for each edge $xy \in E(G)$. The complete graphs are the only known examples of double-critical graphs. A long-standing…

Halin [1965] proved that if a graph has $n$ many pairwise disjoint rays for each $n$ then it has infinitely many pairwise disjoint rays. We analyze the complexity of this and other similar results in terms of computable and proof theoretic…

Paper withdrawn by the author.

We deal with $r$-regular bipartite graphs with $2n$ vertices. In a previous paper, Butera, Pernici and the author have introduced a quantity $u(i)$, $u(i) = -\ln(i!m\_i)$, a function of the number of $i$-matchings, $m\_i$, and conjectured…

We obtain heat kernel estimates for a class of fourth order non-uniformly elliptic operators in two dimensions. Contrary to existing results, the operators considered have symbols that are not strongly convex. This rises certain…

We consider changes in properties of a subgraph of an infinite graph resulting from the addition of open edges of Bernoulli percolation on the infinite graph to the subgraph. We give the triplet of an infinite graph, one of its subgraphs,…

We show that, on a complete, connected and non-compact Riemannian manifold of non-negative Ricci curvature, the solution to the heat equation with $L^{1}$ initial data behaves asymptotically as the mass times the heat kernel. In contrast to…