Related papers: On the generalized Tur\'an problem for odd cycles

If K is an odd-dimensional flag closed manifold, flag generalized homology sphere or a more general flag weak pseudomanifold with sufficiently many vertices, then the maximal number of edges in K is achieved by the balanced join of cycles.…

We show that every $n$-vertex $5$-connected planar triangulation has at most $9n-50$ many cycles of length $5$ for all $n\ge 20$ and this upper bound is tight. We also show that for every $k\geq 6$, there exists some constant $C(k)$ such…

Let $G$ be a 2-connected $n$-vertex graph and $N_s(G)$ be the total number of $s$-cliques in $G$. Let $k\ge 4$ and $s\ge 2$ be integers. In this paper, we show that if $G$ has an edge $e$ which is not on any cycle of length at least $k$,…

For a graph $H$, the Tur\'{a}n number of $H$, denoted by ex$(n,H)$, is the maximum number of edges of an $n$-vertex $H$-free graph. Let $g(n,H)$ denote the maximum number of edges not contained in any monochromatic copy of $H$ in a…

We show that every graph $G$ on $n$ vertices with $\delta(G) \geq (1/2+\varepsilon)n$ is spanned by a complete blow-up of a cycle with clusters of nearly uniform size $\Omega(\log n)$. The proof is based on a recently introduced approach…

Let $G$ be a graph, and $v\in V(G)$ and $S\subseteq V(G)\backslash v$ of size at least $k$. An important result on graph connectivity due to Perfect states that, if $v$ and $S$ are $k$-linked, then a $(k-1)$-link between a vertex $v$ and…

A conjecture of Verstra\"ete states that for any fixed $\ell < k$ there exists a positive constant $c$ such that any $C_{2k}$-free graph $G$ contains a $C_{2\ell}$-free subgraph with at least $c |E(G)|$ edges. For $\ell = 2$, this…

It is shown that for $n\geq 141$, among all triangle-free graphs on $n$ vertices, the complete equibipartite graph is the unique triangle-free graph with the greatest number of cycles.

More than forty years ago, Erd\H{o}s conjectured that for any T <= N/K, every K-uniform hypergraph on N vertices without T disjoint edges has at most max{\binom{KT-1}{K}, \binom{N}{K} - \binom{N-T+1}{K}} edges. Although this appears to be a…

We address a problem which is a generalization of Tur\'an-type problems recently introduced by Imolay, Karl, Nagy and V\'ali. Let $F$ be a fixed graph and let $G$ be the union of $k$ edge-disjoint copies of $F$, namely $G =…

In 1979, Hakimi and Schmeichel considered the problem of maximizing the number of cycles of a given length in an $n$-vertex planar graph. They precisely determined the maximum number of triangles and $4$-cycles and presented a conjecture…

We describe the C_{2k+1}-free graphs on n vertices with maximum number of edges. The extremal graphs are unique except for n = 3k-1, 3k, 4k-2, or 4k-1. The value of ex(n,C_{2k+1}) can be read out from the works of Bondy, Woodall, and…

Let $\mathbf{k} := (k_1,\dots,k_s)$ be a sequence of natural numbers. For a graph $G$, let $F(G;\mathbf{k})$ denote the number of colourings of the edges of $G$ with colours $1,\dots,s$ such that, for every $c \in \{1,\dots,s\}$, the edges…

We prove that for all natural numbers $m$ and $k$ where $k$ is odd, there exists a natural number $N(k)$ such that any 3-connected cubic graph with at least $N(k)$ vertices contains a cycle of length $m$ modulo $k$. We also construct a…

The Tur\'an number of a $k$-uniform hypergraph $H$, denoted by $e{x_k}\left({n;H} \right)$, is the maximum number of edges in any $k$-uniform hypergraph $F$ on $n$ vertices which does not contain $H$ as a subgraph. Let…

For any positive integer $k$, we show that every maximal $C_{2k+1}$-free graph with at least $n^2/4-o(n^{3/2})$ edges contains an induced complete bipartite subgraph on $(1-o(1))n$ vertices. We also show that this is best possible.

Consider the family of graphs without $ k $ node-disjoint odd cycles, where $ k $ is a constant. Determining the complexity of the stable set problem for such graphs $ G $ is a long-standing problem. We give a polynomial-time algorithm for…

For any $k\ge 3$ and $\ell \in [k-1]$ such that $(k,\ell) \ne (3,1)$, we show that any sufficiently large $k$-graph $G$ must contain a Hamilton $\ell$-cycle provided that it has no isolated vertices and every set of $k-1$ vertices contained…

An $acyclic$ edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The \emph{acyclic chromatic index} of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and…

The expansion $F^{\triangle}$ of a graph $F$ is the graph obtained from $F$ by replacing each edge with a triangle. Lv \etal proposed a conjecture on the maximum number of triangles in a graph without $P_k^{\triangle}$ or $C_k^{\triangle}$…