Related papers: Stars on trees

A subset of vertices is a {\it maximum independent set} if no two of the vertices are adjacent and the subset has maximum cardinality. A subset of vertices is called a {\it maximum dissociation set} if it induces a subgraph with vertex…

Given a positive integer $r$ and a graph $G$ with degree sequence $d_1,\dots,d_n$, we define $e_r(G)=\sum_{i=1}^n d_i^r$. We let $\mathrm{ex}_r(n,F)$ be the largest value of $e_r(G)$ if $G$ is an $n$-vertex $F$-free graph. We show that if…

For a graph $G$, denote by $t_r(G)$ (resp. $b_r(G)$) the maximum size of a $K_r$-free (resp. $(r-1)$-partite) subgraph of $G$. Of course $t_r(G) \geq b_r(G)$ for any $G$, and Tur\'an's Theorem says that equality holds for complete graphs.…

Let $\alpha(G)$ denote the cardinality of a maximum independent set, while $\mu(G)$ be the size of a maximum matching in the graph $G=\left(V,E\right) $. If $\alpha(G)+\mu(G)=\left\vert V\right\vert $, then $G$ is a K\"onig-Egerv\'ary…

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…

Nikiforov (LAA, 2010) conjectured that for given integer $k$, any graph $G$ of sufficiently large order $n$ with spectral radius $\mu(G)\geq \mu(S_{n,k})$ contains all trees of order $2k+2$, unless $G=S_{n,k}$, where $S_{n,k}=K_k\vee…

The $k$th power of a graph $G$, denoted $G^k$, has the same vertex set as $G$, and two vertices are adjacent in $G^k$ if and only if there exists a path between them in $G$ of length at most $k$. A $K_r$-factor in a graph is a spanning…

For an integer $k$ at least $2$, and a graph $G$, let $f_k(G)$ be the minimum cardinality of a set $X$ of vertices of $G$ such that $G-X$ has either $k$ vertices of maximum degree or order less than $k$. Caro and Yuster (Discrete…

Aboulker, Adler, Kim, Sintiari, and Trotignon conjectured that every graph with bounded maximum degree and large treewidth must contain, as an induced subgraph, a large subdivided wall, or the line graph of a large subdivided wall. This…

Let $G$ be a graph and let $f$ be a positive integer-valued function on $V(G)$ satisfying $2m\le f\le b$, where $b$ and $m$ are two positive integers with $b\ge 4m^2$. In this paper, we show that if $G$ is $b^2$-tough and $|V(G)|\ge b^2$,…

Given a spanning tree $T$ of a planar graph $G$, the co-tree of $T$ is the spanning tree of the dual graph $G^*$ with edge set $(E(G)-E(T))^*$. Gr\"unbaum conjectured in 1970 that every planar 3-connected graph $G$ contains a spanning tree…

A widely open conjecture proposed by Bollob\'as, Erd\H{o}s, and Tuza in the early 1990s states that for any $n$-vertex graph $G$, if the independence number $\alpha(G) = \Omega(n)$, then there is a subset $T \subseteq V(G)$ with $|T| =…

Let $H$ be a graph. We show that if $r$ is large enough as a function of $H$, then the $r$-partite Tur\'an graph maximizes the number of copies of $H$ among all $K_{r+1}$-free graphs on a given number of vertices. This confirms a conjecture…

A classical result of Koml\'os, S\'ark\"ozy and Szemer\'edi states that every $n$-vertex graph with minimum degree at least $(1/2+ o(1))n$ contains every $n$-vertex tree with maximum degree $O(n/\log{n})$ as a subgraph, and the bounds on…

Given two non-empty graphs $H$ and $T$, write $H\succcurlyeq T$ to mean that $t(H,G)^{|E(T)|}\geq t(T,G)^{|E(H)|}$ for every graph $G$, where $t(\cdot,\cdot)$ is the homomorphism density function. We obtain various necessary and sufficient…

For a graph \(G\) with no isolated vertices, its Laplacian ratio is defined as \[ \pi(G)=\frac{\operatorname{per}(L(G))}{\prod_{v\in V(G)} d(v)}, \] where \(L(G)\) is the Laplacian matrix of \(G\), \(d(v)\) is the degree of \(v\), and…

Extremal problems concerning the number of independent sets or complete subgraphs in a graph have been well studied in recent years. Cutler and Radcliffe proved that among graphs with $n$ vertices and maximum degree at most $r$, where $n =…

Let $G$ be a graph on $n$ vertices. A linear forest is a graph consisting of vertex-disjoint paths and isolated vertices. A maximum linear forest of $G$ is a subgraph of $G$ with maximum number of edges, which is a linear forest. We denote…

Let $T$ be a tree with $t$ edges. We show that the number of isomorphic (labeled) copies of $T$ in a graph $G = (V,E)$ of minimum degree at least $t$ is at least \[2|E| \prod_{v \in V} (d(v) - t + 1)^{\frac{(t-1)d(v)}{2|E|}}.\]…

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…