Related papers: All longest cycles intersect in partial 3-trees
The {\em circumference} of a graph $G$ with at least one cycle is the length of a longest cycle in $G$. A classic result of Birmel\'e (2003) states that the treewidth of $G$ is at most its circumference minus $1$. In case $G$ is…
It is shown that for $n\geq 141$, among all triangle-free graphs on $n$ vertices, the complete equibipartite graph is the unique triangle-free graph with the greatest number of cycles.
We prove that a complete bipartite graph can be decomposed into cycles of arbitrary specified lengths provided that the obvious necessary conditions are satisfied, the length of each cycle is at most the size of the smallest part, and the…
In this review I discuss intersection numbers of twisted cocycles and their relation to physics. After defining what these intersection number are, I will first discuss a method for computing them. This is followed by three examples where…
We show that the fact that a political decision filtered through a finite tree of committees gives a determined answer generalises in some sense to infinite trees. This implies a new special case of the Matroid Intersection Conjecture.
Phylogenetic reconciliation seeks to explain host-symbiont co-evolution by mapping parasite trees onto host trees through events such as cospeciation, duplication, host switching, and loss. Finding an optimal reconciliation that ensures…
In the interchange process on a graph $G=(V,E)$, distinguished particles are placed on the vertices of $G$ with independent Poisson clocks on the edges. When the clock of an edge rings, the two particles on the two sides of the edge…
We establish necessary and sufficient conditions for the existence of a decomposition of a complete multigraph into edge-disjoint cycles of specified lengths, or into edge-disjoint cycles of specified lengths and a perfect matching.
We solve a recent question of Caro, Patk\'os and Tuza by determining the exact maximum number of edges in a bipartite connected graph as a function of the longest path it contains as a subgraph and of the number of vertices in each side of…
We determine all 2- and 3-cocycles for Laver tables, an infinite sequence of finite structures obeying the left-selfdistributivity law; in particular, we describe simple explicit bases. This provides a number of new positive braid…
In this paper we give a new and simplified proof of the variational Hodge conjecture for complete intersection cycles on a hypersurface in projective space.
We study the lattices of algebraic and transcendental cycles of cubic fourfolds.
We demonstrate formation of hierarchical structures in two-dimensional systems with multiple length scales in the inter-particle interaction. These include states such as clusters of clusters, concentric rings, clusters inside a ring, and…
We first prove a one-to-one correspondence between finding Hamiltonian cycles in a cubic planar graphs and finding trees with specific properties in dual graphs. Using this information, we construct an exact algorithm for finding…
It implicitly follows from the work of [Colbourn, El-Mallah: On two dual classes of planar graphs. Discrete Mathematics 80(1): 21-40 (1990)] that every planar partial 3-tree is a subgraph of a planar 3-tree. This fact has already enabled to…
Let $G$ be a $k$-connected graph on $n$ vertices. Hippchen's Conjecture states that two longest paths in $G$ share at least $k$ vertices. Guti\'errez recently proved the conjecture when $k\leq 4$ or $k\geq \frac{n-2}{3}$. We improve upon…
A family of $k$-element subsets of an $n$-element set is called 3-wise intersecting if any three members in the family have non-empty intersection. We determine the maximum size of such families exactly or asymptotically. One of our results…
Luo, Tian and Wu conjectured in 2022 that for any tree $T$ with bipartition $X$ and $Y$, every $k$-connected bipartite graph $G$ with $\delta(G) \geq k + t$, where $t = \max\{|X|,|Y |\}$, contains a subtree $T' \cong T$ such that $G-V(T')$…
We study the size and structure of the largest common subtree (LCS) between two independent Bienaym\'e trees conditioned to have size $n$. When the trees are critical with finite $2$nd and $(2+\kappa)$th moment respectively for some…
Which $2$-regular subgraph $R$ of a cubic graph $G$ can be extended to a cycle double cover of $G$? We provide a condition which ensures that every $R$ satisfying this condition is part of a cycle double cover of $G$. As one consequence, we…