Related papers: Super-minimally $3$-connected graphs

A theorem of Tutte states that every 4-connected non-hamiltonian graph contains $K_{3,3}$ as a minor. We strengthen this result by proving that such a graph must contain $K_{3,4}$ as a minor, thereby confirming a special case of a…

A connected graph $\G$ is said to be {\it distance-balanced} whenever for any pair of adjacent vertices $u,v$ of $\G$ the number of vertices closer to $u$ than to $v$ is equal to the number of vertices closer to $v$ than to $u$. In…

An edge subset \( S \subseteq E(G) \) is called a 3-restricted edge-cut if \( G - S \) is disconnected and each component of \( G - S \) contains at least three vertices. The 3-restricted edge-connectivity of a graph \( G \), denoted by \(…

A graph $G$ is a non-separating planar graph if there is a drawing $D$ of $G$ on the plane such that (1) no two edges cross each other in $D$ and (2) for any cycle $C$ in $D$, any two vertices not in $C$ are on the same side of $C$ in $D$.…

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 $G-V(T')$ is $k$-connected. In this paper, we give a…

For graphs $G$ and $H$, an $H$-coloring of $G$ is a map from the vertices of $G$ to the vertices of $H$ that preserves edge adjacency. We consider the following extremal enumerative question: for a given $H$, which connected $n$-vertex…

A graph G is called (2k, k)-connected if G is 2k-edge-connected and G-v is k-edge-connected for every vertex v. The study of (2k, k)-connected graphs is motivated by a conjecture of Frank which states that a graph has a 2-vertex-connected…

In this paper, we classify all the finite groups $G$ such that the commuting graph $\Gamma_C(G)$, order-sum graph $\Gamma_{OS}(G)$ and non-inverse graph $\Gamma_{NI}(G)$ are minimally edge connected graphs. We also classify all the finite…

We prove that every connected cubic graph with $n$ vertices has a maximal matching of size at most $\frac{5}{12} n+ \frac{1}{2}$. This confirms the cubic case of a conjecture of Baste, F\"urst, Henning, Mohr and Rautenbach (2019) on regular…

We call a finite undirected graph minimally k-matchable if it has at least k distinct perfect matchings but deleting any edge results in a graph which has not. An odd subdivision of some graph G is any graph obtained by replacing every edge…

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…

We study the problem of maximizing the number of independent sets in $n$-vertex $k$-chromatic $\ell$-connected graphs. First we consider maximizing the total number of independent sets in such graphs with $n$ sufficiently large, and for…

A graph $G$ is $\mathcal S_3$-connected if, for any mapping $\beta : V (G) \mapsto {\mathbb Z}_3$ with $\sum_{v\in V(G)} \beta(v)\equiv 0\pmod3$, there exists a strongly connected orientation $D$ satisfying $d^{+}_D(v)-d^{-}_D(v)\equiv…

An edge of a quasi $k$-connected graph is said to be quasi $k$-contractible if the contraction of the edge results in a quasi $k$-connected graph. We show that every 5-connected graph contains a quasi 5-contractible edge. Furthermore, we…

We determine a lower bound for the number of edges of a 2-connected maximal nontraceable graph, and present a construction of an infinite family of maximal nontraceable graphs that realize this bound.

The concept of pendant-tree $k$-connectivity $\tau_k(G)$ of a graph $G$, introduced by Hager in 1985, is a generalization of classical vertex-connectivity. Let $f(n,k,\ell)$ be the minimal number of edges of a graph $G$ of order $n$ with…

Let $G$ be a nontrivial connected graph of order $n$ and let $k$ be an integer with $2\leq k\leq n$. For a set $S$ of $k$ vertices of $G$, let $\kappa (S)$ denote the maximum number $\ell$ of edge-disjoint trees $T_1,T_2,...,T_\ell$ in $G$…

Let $n,k,s$ be three integers and $\beta$ be a sufficiently small positive number such that $k\geq 3$, $0<1/n\ll \beta\ll 1/k$ and $ks+k\leq n\leq (1+\beta)ks+k-2$. A $k$-graph is called non-trivial if it has no isolated vertex. In this…

In this paper we raise a variant of a classic problem in extremal graph theory, which is motivated by a design of fractional repetition codes, a model in distributed storage systems. For any feasible positive integers $d\geq 3$, $n \geq 3$,…

For a graph $G$, let $c_k(G)$ be the number of spanning trees of $G$ with maximum degree at most $k$. For $k \ge 3$, it is proved that every connected $n$-vertex $r$-regular graph $G$ with $r \ge \frac{n}{k+1}$ satisfies $$ c_k(G)^{1/n} \ge…