Related papers: Mader's conjecture for graphs with small connectiv…

The class of cographs is one of the most well-known graph classes, which is also known to be equivalent to the class of $P_4$-free graphs. We show that Mader's conjecture is true if we restrict ourselves to cographs, that is, for any tree…

W. Mader [J. Graph Theory 65 (2010), 61--69] conjectured that for any tree $T$ of order $m$, every $k$-connected graph $G$ with $\delta(G)\geq\lfloor\frac{3k}{2}\rfloor+m-1$ contains a tree $T'\cong T$ such that $G-V(T')$ remains…

Mader [J. Combin. Theory Ser. B 40 (1986) 152-158] proved that every $k$-edge-connected graph $G$ with minimum degree at least $k+1$ contains a vertex $u$ such that $G-\{u\}$ is still $k$-edge-connected. In this paper, we prove that every…

In 2012, Mader conjectured that for any tree $T$ of order $m$, every $k$-connected graph $G$ with minimum degree at least $\lfloor \frac{3k}{2}\rfloor+m-1$ contains a subtree $T'\cong T$ such that $G-V(T')$ remains $k$-connected. In 2022,…

In [W. Mader, Connectivity keeping paths in $k$-connected graphs, J. Graph Theory 65 (2010) 61-69.], Mader conjectured that for every positive integer $k$ and every finite tree $T$ with order $m$, every $k$-connected, finite graph $G$ with…

Mader [J. Graph Theory 65 (2010) 61-69] conjectured that for every positive integer $k$ and every finite tree $T$ with order $m$, every $k$-connected, finite graph $G$ with $\delta(G)\geq \lfloor\frac{3}{2}k\rfloor+m-1$ contains a subtree…

Hasunuma [J. Graph Theory 102 (2023) 423-435] conjectured that for any tree $T$ of order $m$, every $k$-connected (or $k$-edge-connected) graph $G$ with minimum degree at least $k+m-1$ contains a tree $T'\cong T$ such that $G-E(T')$ is…

In 2010, Mader [W. Mader, Connectivity keeping paths in $k$-connected graphs, J. Graph Theory 65 (2010) 61-69.] proved that every $k$-connected graph $G$ with minimum degree at least $\lfloor\frac{3k}{2}\rfloor+m-1$ contains a path $P$ of…

Luo, Tian and Wu (2022) conjectured that for any tree $T$ with bipartition $X$ and $Y$, every $k$-connected bipartite graph $G$ with minimum degree at least $k+t$, where $t=$max$\{|X|,|Y|\}$, contains a tree $T'\cong T$ such that $G-V(T')$…

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 a subgraph. Our result confirms, for…

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…

Mader conjectured in 1979 that an average degree of at least $3k-1$ in a graph is sufficient for the existence of a $(k+1)$-connected subgraph. The following minimum degree analogue holds: Every graph with minimum degree at least $3k-1$…

A variant of the Erd\H{o}s-S\'os conjecture, posed by Havet, Reed, Stein and Wood, states that every graph with minimum degree at least $\lfloor 2k/3 \rfloor$ and maximum degree at least $k$ contains a copy of every tree with $k$ edges.…

Mader proved that for $k\geq 2$ and $n\geq 2k$, every $n$-vertex graph with no $(k+1)$-connected subgraphs has at most $(1+\frac{1}{\sqrt{2}})k(n-k)$ edges. He also conjectured that for $n$ large with respect to $k$, every such graph has at…

Luo, Tian and Wu [Discrete Math. 345 (4) (2022) 112788] conjectured that for any tree $T$ with bipartition $(X,Y)$, every $k$-connected bipartite graph $G$ with minimum degree at least $k+w$, where $w=\max\{|X|,|Y|\}$, contains a tree…

A fundamental result of Mader from 1972 asserts that a graph of high average degree contains a highly connected subgraph with roughly the same average degree. We prove a lemma showing that one can strengthen Mader's result by replacing the…

The Erd\H{o}s-S\'os Conjecture states that every graph with average degree exceeding $k-1$ contains every tree with $k$ edges as a subgraph. We prove that there are $\delta>0$ and $k_0\in\mathbb N$ such that the conjecture holds for every…

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…

A k-tree is either a complete graph on (k+1) vertices or given a k-tree G' with n vertices, a k-tree G with (n+1) vertices can be constructed by introducing a new vertex v and picking a k-clique Q in G' and then joining each vertex u in Q.…

Let $G=(V,E)$ be a complete $n$-vertex graph with distinct positive edge weights. We prove that for $k\in\{1,2,...,n-1\}$, the set consisting of the edges of all minimum spanning trees (MSTs) over induced subgraphs of $G$ with $n-k+1$…