Related papers: A generalization of the Chv\'{a}tal-Erd\H{o}s theo…

The celebrated K\H{o}v\'ari-S\'os-Tur\'an theorem states that any $n$-vertex graph containing no copy of the complete bipartite graph $K_{s,s}$ has at most $O_s(n^{2-1/s})$ edges. In the past two decades, motivated by the applications in…

The generalized connectivity of a graph, which was introduced recently by Chartrand et al., is a generalization of the concept of vertex connectivity. Let $S$ be a nonempty set of vertices of $G$, a collection $\{T_1,T_2,...,T_r\}$ of trees…

The famous K\H{o}nig-Egerv\'ary theorem is equivalent to the statement that the matching number equals the vertex cover number for every induced subgraph of some graph if and only if that graph is bipartite. Inspired by this result, we…

An ordered graph is a graph with a linear ordering on its vertex set. We prove that for every positive integer $k$, there exists a constant $c_k>0$ such that any ordered graph $G$ on $n$ vertices with the property that neither $G$ nor its…

For a given graph $H$, a graph $G$ is $H$-linked if, for every injection $\varphi: V(H) \to V(G)$, the graph $G$ contains a subdivision of $H$ with $\varphi(v)$ corresponding to $v$, for each $v\in V(H)$. Let $f(H)$ be the minimum integer…

- A hamiltonian graph $G$ verifying $e(G)>n(k-1)/2$ %with a vertex of degree greater or equal than $k$ contains any $k$-spider. - If $G$ is a graph with average degree $\bar{d} > k-1$, then every spider of size $k$ is contained in $G$ for…

Erd\H{o}s and Hajnal conjectured that for every graph $H$, there exists $c>0$ such that every $H$-free graph $G$ has a clique or a stable set of size at least $|G|^c$ (a graph is $H$-free if it has no induced subgraph isomorphic to $H$).…

The symmetric difference of two graphs $G_1,G_2$ on the same set of vertices $V$ is the graph on $V$ whose set of edges are all edges that belong to exactly one of the two graphs $G_1,G_2$. For a fixed graph $H$ call a collection ${\cal G}$…

A graph is $\mathcal{R}_d$-independent (resp. $\mathcal{R}_d$-connected) if its $d$-dimensional generic rigidity matroid is free (resp. connected). A result of Maxwell from 1867 implies that every $\mathcal{R}_d$-independent graph satisfies…

The conditional $h$-vertex($h$-edge) connectivity of a connected graph $H$ of minimum degree $ k > h$ is the size of a smallest vertex(edge) set $F$ of $H$ such that $H - F$ is a disconnected graph of minimum degree at least $h.$ Let $G$ be…

A $k$-container $C(u, v)$ of a graph $G$ is a set of $k$ internally disjoint paths between $u$ and $v$. A $k$-container $C(u, v)$ of $G$ is a $k^*$-container if it is a spanning subgraph of $G$. A graph $G$ is $k^*$-connected if there…

Let $G=(V(G), E(G))$ be a graph with vertex set $V(G)$ and edge set $E(G)$. For $k\geq2$ and given any subset $S\subseteq|V(G)|$ with $|S|=k$, if a graph $G$ of order $|V(G)|\geq k+1$ always has a spanning tree $T$ such that $S$ is…

For a graph $G=(V,E)$ and a set $S\subseteq V(G)$ of size at least $2$, a path in $G$ is said to be an $S$-path if it connects all vertices of $S$. Two $S$-paths $P_1$ and $P_2$ are said to be internally disjoint if $E(P_1)\cap…

Let $G$ be a (multi)graph of order $n$ and let $u,v$ be vertices of $G$. The maximum number of internally disjoint $u$-$v$ paths in $G$ is denoted by $\kappa_G(u,v)$, and the maximum number of edge-disjoint $u$-$v$ paths in $G$ is denoted…

For a non-negative integer $s\le |V(G)|-3$, a graph $G$ is $s$-Hamiltonian if the removal of any $k\le s$ vertices results in a Hamiltonian graph. Given a connected simple graph $G$ that is not isomorphic to a path, a cycle, or a $K_{1,3}$,…

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 a graph $G$ with at least two vertices, the maximum local edge-connectivity of $G$ is the maximum number of edge-disjoint $(u,v)$-paths over all distinct pairs of vertices $(u,v)$ in $G$. Stiebitz and Toft (2018) proved a Brooks-type…

The inducibility of a graph $H$ measures the maximum number of induced copies of $H$ a large graph $G$ can have. Generalizing this notion, we study how many induced subgraphs of fixed order $k$ and size $\ell$ a large graph $G$ on $n$…

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…

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…