相关论文: On the Turan number of forests

Given a family ${\cal F}$ of graphs, and a positive integer $n$, the Tur\'an number $ex(n,{\cal F})$ of ${\cal F}$ is the maximum number of edges in an $n$-vertex graph that does not contain any member of ${\cal F}$ as a subgraph. The order…

Given graphs $T$ and $H$, the generalized Tur\'an number ex$(n,T,H)$ is the maximum number of copies of $T$ in an $n$-vertex graph with no copies of $H$. Alon and Shikhelman, using a result of Erd\H os, determined the asymptotics of…

The classic extremal problem is that of computing the maximum number of edges in an $F$-free graph. In the case where $F=K_{r+1}$, the extremal number was determined by Tur\'an. Later results, known as supersaturation theorems, proved that…

We study $\mathrm{exa}_k(n,F)$, the largest number of edges in an $n$-vertex graph $G$ that contains exactly $k$ copies of a given subgraph $F$. The case $k=0$ is the Tur\'an number $\mathrm{ex}(n,F)$ that is among the most studied…

Let $\mathrm{rex}(n, F)$ denote the maximum number of edges in an $n$-vertex graph that is regular and does not contain $F$ as a subgraph. We give lower bounds on $\mathrm{rex}(n, F)$, that are best possible up to a constant factor, when…

Given a graph $T$ and a family of graphs $\mathcal{F}$, the generalized Tur\'an number of $\mathcal{F}$ is the maximum number of copies of $T$ in an $\mathcal{F}$-free graph on $n$ vertices, denoted by $ex(n,T,\mathcal{F})$. When $T = K_r$,…

Let $k\ge 2$ and $n_1\ge n_2\ge n_3\ge n_4$ be integers such that $n_4$ is sufficiently larger than $k$. We determine the maximum number of edges of a 4-partite graph with parts of sizes $n_1,\dots, n_4$ that does not contain $k$…

Given a graph $H$ and an odd integer $t$ ($t\geq 3$), the odd-ballooning of $H$, denoted by $H(t)$, is the graph obtained from replacing each edge of $H$ by an odd cycle of length at least $t$ where the new vertices of the cycles are all…

A non-uniform hypergraph $H=(V,E)$ consists of a vertex set $V$ and an edge set $E\subseteq 2^V$; the edges in $E$ are not required to all have the same cardinality. The set of all cardinalities of edges in $H$ is denoted by $R(H)$, the set…

Given graphs $H$ and $F$, the generalized Tur\'an number $\mathrm{ex}(n,H,F)$ is the maximum number of copies of $H$ among all $n$-vertex $F$-free graphs. The friendship graph $F_k$ consists of $k$ triangles sharing a common vertex. In this…

Let $\mathcal{F}$ be a family of $r$-uniform hypergraphs. The random Tur\'an number $\mathrm{ex}(G^r_{n,p},\mathcal{F})$ is the maximum number of edges in an $\mathcal{F}$-free subgraph of $G^r_{n,p}$, where $G^r_{n,p}$ is the…

Given $p\geq 0$ and a graph $G$ whose degree sequence is $d_1,d_2,\ldots,d_n$, let $e_p(G)=\sum_{i=1}^n d_i^p$. Caro and Yuster introduced a Tur\'an-type problem for $e_p(G)$: given $p\geq 0$, how large can $e_p(G)$ be if $G$ has no…

For a graph $G$ whose degree sequence is $d_{1},..., d_{n}$, and for a positive integer $p$, let $e_{p}(G)=\sum_{i=1}^{n}d_{i}^{p}$. For a fixed graph $H$, let $t_{p}(n,H)$ denote the maximum value of $e_{p}(G)$ taken over all graphs with…

The planar Tur\'a number of a graph $F$ is the maximum number of edges an $n$-vertex $F$-free planar graph can have. We study the case where $F$ is forbidden as an induced subgraph, thereby introducing the induced planar Tur\'a numbers. We…

Fix a $k$-chromatic graph $F$. In this paper we consider the question to determine for which graphs $H$ does the Tur\'an graph $T_{k-1}(n)$ have the maximum number of copies of $H$ among all $n$-vertex $F$-free graphs (for $n$ large…

For graphs $H$ and $F$, the generalized Tur\'an number $ex(n,H,F)$ is the largest number of copies of $H$ in an $F$-free graph on $n$ vertices. We consider this problem when both $H$ and $F$ have at most four vertices. We give sharp results…

The expansion $G^+$ of a graph $G$ is the $3$-uniform hypergraph obtained from $G$ by enlarging each edge of $G$ with a new vertex disjoint from $V(G)$ such that distinct edges are enlarged by distinct vertices. Let $ex_3(n,F)$ denote the…

We suggest a new type of problem about distances in graphs and make several conjectures. As a first step towards proving them, we show that for sufficiently large values of n and k, a graph on n vertices that has no three vertices at…

Hajos' conjecture that every simple even graph on $n$ vertices can be decomposed into at most $(n-1)/2$ cycles (see L. Lovasz, On covering of graphs, in: P. Erdos, G.O.H. Katona (Eds.), Theory of Graphs, Academic Press, New York, 1968, pp.…

Given two graphs $G$ and $H$ with $H\subseteq G$ we consider the anti-Ramsey function $AR(G,H)$ which is the maximum number of colors in any edge-coloring of $G$ so that every copy of $H$ receives the same color on at least one pair of…