Related papers: An elementary proof of the bunkbed conjecture for …
An induced forest of a graph G is an acyclic induced subgraph of G. The present paper is devoted to the analysis of a simple randomised algorithm that grows an induced forest in a regular graph. The expected size of the forest it outputs…
A conjecture of Barnette states that every 3-connected cubic bipartite plane graph has a Hamilton cycle, which is equivalent to the statement that every simple even plane triangulation admits a partition of its vertex set into two subsets…
The Gy\'arf\'as tree packing conjecture asserts that any set of trees with $2,3, ..., k$ vertices has an (edge-disjoint) packing into the complete graph on $k$ vertices. Gy\'arf\'as and Lehel proved that the conjecture holds in some special…
We give a short elementary proof of Tutte and Nash-Williams' characterization of graphs with k edge-disjoint spanning trees.
For \math{p\ge 1}, we prove that every forest with \math{p} trees whose sizes are $a_1,..., a_p$ can be embedded in any graph containing at least $\sum_{i=1}^p (a_i + 1)$ vertices and having a minimum degree at least $\sum_{i=1}^p a_i$.
A linear forest is a forest in which every connected component is a path. The linear arboricity of a graph $G$ is the minimum number of linear forests of $G$ covering all edges. In 1980, Akiyama, Exoo and Harary proposed a conjecture, known…
In this paper a Bass-Serre theory in the groupoid setting is developed and a structure theorem is established. Any groupoid action without inversion of edges on a forest induces a graph of groupoids, while any graph of groupoids satisfying…
The bunkbed of a graph $G$ is the graph $G\times\left\{ 0,1\right\} $. It has been conjectured that in the independent bond percolation model, the probability for $\left(u,0\right)$ to be connected with $\left(v,0\right)$ is greater than…
We study a problem on edge percolation on product graphs $G\times K_2$. Here $G$ is any finite graph and $K_2$ consists of two vertices $\{0,1\}$ connected by an edge. Every edge in $G\times K_2$ is present with probability $p$ independent…
We propose the conjecture that every graph $G$ of order $n$ with less than $3n-6$ edges has a vertex cut that induces a forest. Maximal planar graphs do not have such vertex cuts and show that the density condition would be best possible.…
The bunkbed conjecture was first posed by Kasteleyn. If $G=(V,E)$ is a finite graph and $H$ some subset of $V$, then the bunkbed of the pair $(G,H)$ is the graph $G\times\{1,2\}$ plus $|H|$ extra edges to connect for every $v\in H$ the…
Konig's theorem states that the covering number and the matching number of a bipartite graph are equal. We prove a generalisation of this result, in which each point in one side of the graph is replaced by a subtree of a given tree. The…
The Ferrers bound conjecture is a natural graph-theoretic extension of the enumeration of spanning trees for Ferrers graphs. We document the current status of the conjecture and provide a further conjecture which implies it.
In this paper and a companion paper, we prove that, if $m$ is sufficiently large, every graph on $m+1$ vertices that has a universal vertex and minimum degree at least $\lfloor \frac{2m}{3} \rfloor$ contains each tree $T$ with $m$ edges as…
We show a general result known as the Erdos_Sos Conjecture: if $E(G)>{1/2}(k-1)n$ where $G$ has order $n$ then $G$ contains every tree of order $k+1$ as a subgraph.
The bunkbed of a graph $G$ is the graph $G\times K_2 $. It has been conjectured that in the independent bond percolation model, the probability for $\left(u,0\right)$ to be connected with $\left(v,0\right)$ is greater than the probability…
A "folklore conjecture, probably due to Tutte" (as described in [P.D. Seymour, Sums of circuits, Graph theory and related topics (Proc. Conf., Univ. Waterloo, 1977), pp. 341-355, Academic Press, 1979]) asserts that every bridgeless cubic…
We give an elementary, self-contained, and purely combinatorial proof of the Rayleigh monotonicity property of graphs.
For a finite simple graph $G$, the bunkbed graph $G^\pm$ is defined to be the product graph $G\square K_2$. We will label the two copies of a vertex $v\in V(G)$ as $v_-$ and $v_+$. The bunkbed conjecture, posed by Kasteleyn, states that for…
The statement in the title was proved in \cite{Cao23} by introducing dominant sets of seeds, which are analogs of torsion classes in representation theory. In this note, we observe a short proof by the existence of consistent cluster…