Related papers: Linkages in Large Graphs of Bounded Tree-Width

Chv\'{a}tal and Erd\"{o}s [Discrete Math. 2 (1972) 111-113] stated that, for an $m$-connected graph $G$, if its independence number $\alpha(G)\leq m-1$, then $G$ is Hamilton-connected. Note that $k$-leaf-connectedness is a natural…

Graphs with bounded treewidth and bounded maximum degree are known to have tree-partitions of bounded width. What can be said if the bounded treewidth assumption is strengthened to bounded pathwidth? We prove that every graph with bounded…

Thomassen's chord conjecture from 1976 states that every longest cycle in a $3$-connected graph has a chord. This is one of the most important unsolved problems in graph theory. Let $H$ be a subgraph of a graph $G$. A vertex $v$ of $H$ is…

The tree of decomposition of a $k$-connected graph by a set $\mathfrak S$ of pairwise independent $k$-vertex cutsets is defined as follows. The vertices of this tree are cutsets of $\mathfrak S$ and parts of decomposition of the graph by…

Let $G$ be a simple graph of order $n\geq 2$ and let $k\in \{1,\ldots ,n-1\}$. The $k$-token graph $F_k(G)$ of $G$ is the graph whose vertices are the $k$-subsets of $V(G)$, where two vertices are adjacent in $F_k(G)$ whenever their…

The Bar\'at-Thomassen conjecture asserts that for every tree $T$ on $m$ edges, there exists a constant $k_T$ such that every $k_T$-edge-connected graph with size divisible by $m$ can be edge-decomposed into copies of $T$. So far this…

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 $G-V(T')$…

Diestel and M\"uller showed that the connected tree-width of a graph $G$, i.e., the minimum width of any tree-decomposition with connected parts, can be bounded in terms of the tree-width of $G$ and the largest length of a geodesic cycle in…

In 1960 Nash-Williams proved that an edge-connectivity of 2k is sufficient for a finite graph to have a k-arc-connected orientation. He then conjectured that the same is true for infinite graphs. In 2016, Thomassen, using his own results on…

For a graph $G=(V,E)$ and a set $S\subseteq V(G)$ of size at least $2$, an $S$-Steiner tree $T$ is a subgraph of $G$ that is a tree with $S\subseteq V(T)$. Two $S$-Steiner trees $T$ and $T'$ are internally disjoint (resp. edge-disjoint) if…

We show that the square of every connected $S(K_{1,4})$-free graph satisfying a matching condition has a $2$-connected spanning subgraph of maximum degree at most~$3$. Furthermore, we characterise trees whose square has a $2$-connected…

We prove that every (6k + 2l, 2k)-connected simple graph contains k rigid and l connected edge-disjoint spanning subgraphs. This implies a theorem of Jackson and Jord\'an [4] and a theorem of Jord\'an [6] on packing of rigid spanning…

For integers $k,n$ with $1 \le k \le n/2$, let $f(k,n)$ be the smallest integer $t$ such that every $t$-connected $n$-vertex graph has a spanning bipartite $k$-connected subgraph. A conjecture of Thomassen asserts that $f(k,n)$ is upper…

Building on recent work by Thomassen, we show that Nash-Williams' orientation theorem, that every finite $2k$-edge-connected multigraph has a $k$-arc-connected orientation, also holds for all infinite multigraphs.

Thomassen's chord conjecture from 1976 states that every longest cycle in a $3$-connected graph has a chord. This is one of the most important unsolved problems in graph theory. We pose a new conjecture which implies Thomassen's conjecture.…

A circle graph is an intersection graph of a set of chords of a circle. We describe the unavoidable induced subgraphs of circle graphs with large treewidth. This includes examples that are far from the `usual suspects'. Our results imply…

A cactus is a connected graph in which each edge is contained in at most one cycle. We generalize the concept of cactus graphs, i.e., a $k$-cactus is a connected graph in which each edge is contained in at most $k$ cycles where $k\ge 1$. It…

We prove that if a graph has a tree-decomposition of width at most w, then it has a tree-decomposition of width at most w with certain desirable properties. We will use this result in a subsequent paper to show that every 2-connected graph…

Carsten Thomassen conjectured that every longest circuit in a 3-connected graph has a chord. We prove the conjecture for graphs having no $K_{3,3}$ minor, and consequently for planar graphs.

Let $G$ be a connected graph and $L(G)$ the set of all integers $k$ such that $G$ contains a spanning tree with exactly $k$ leaves. We show that for a connected graph $G$, the set $L(G)$ is contiguous. It follows from work of Chen, Ren, and…