相关论文: Independent sets in the union of two Hamiltonian c…

Let m(G) denote the number of maximal independent sets of vertices in a graph G and let c(n,r) be the maximum value of m(G) over all connected graphs with n vertices and at most r cycles. A theorem of Griggs, Grinstead, and Guichard gives a…

For integers $k\geq 1$ and $n\geq 2k+1$, the Schrijver graph $S(n,k)$ has as vertices all $k$-element subsets of $[n]:=\{1,2,\ldots,n\}$ that contain no two cyclically adjacent elements, and an edge between any two disjoint sets. More…

A graph construction that produces a k-regular graph on n vertices for any choice of k >= 3 and n = m(k+1) for integer m >= 2 is described. The number of Hamiltonian cycles in such graphs can be explicitly determined as a function of n and…

We say that two graphs $H_1,H_2$ on the same vertex set are $G$-creating ($G$-different in other papers, this difference is explained in the introduction) if the union of the two graphs contains $G$ as a subgraph. Let $H(n,k)$ be the…

An $n$-vertex, $d$-regular graph can have at most $2^{n/2+o_d(n)}$ independent sets. In this paper we address what happens with this upper bound when we impose the further condition that the graph has independence number at most $\alpha$.…

For integers $k\geq 1$ and $n\geq 2k+1$ the Kneser graph $K(n,k)$ has as vertices all $k$-element subsets of $[n]:=\{1,2,\ldots,n\}$ and an edge between any two vertices (=sets) that are disjoint. The bipartite Kneser graph $H(n,k)$ has as…

We study the number of edge-disjoint Hamilton cycles one can guarantee in a sufficiently large graph G on n vertices with minimum degree d = (1/2+a)n. For any constant a > 0, we give an optimal answer in the following sense: let…

For integers $k\geq 1$ and $n\geq 2k+1$, the Kneser graph $K(n,k)$ has as vertices all $k$-element subsets of an $n$-element ground set, and an edge between any two disjoint sets. It has been conjectured since the 1970s that all Kneser…

For a graph $G$, a vertex subset $S$ is called a maximum generalized $k$-independent set if the induced subgraph $G[S]$ does not contain a $k$-tree as its subgraph, and the subset has maximum cardinality. The generalized $k$-independence…

For $k,n\in \mathbb{N}$, the Kneser graph $K(n,k)$ is the graph with vertex set $V=[n]^{(k)}$ and edge set $E=\{\{x,y\} \in V^{(2)}: x\cap y=\emptyset\}$. Chen proved that for $n\geq 3k$, Kneser graphs are Hamiltonian. Similarly as for…

A $k$-uniform hypergraph with $n$ vertices is an $(n,k,\ell)$-omitting system if it does not contain two edges whose intersection has size exactly $\ell$. If in addition it does not contain two edges whose intersection has size greater than…

Let $L$ be a set of positive integers. We call a (directed) graph $G$ an $L$\emph{-cycle graph} if all cycle lengths in $G$ belong to $L$. Let $c(L,n)$ be the maximum number of cycles possible in an $n$-vertex $L$-cycle graph (we use…

Nielsen proved that the maximum number of maximal independent sets (MIS's) of size $k$ in an $n$-vertex graph is asymptotic to $(n/k)^k$, with the extremal construction a disjoint union of $k$ cliques with sizes as close to $n/k$ as…

Fix $k \ge 2$ and let $H$ be a graph with $\chi(H) = k+1$ containing a critical edge. We show that for sufficiently large $n$, the unique $n$-vertex $H$-free graph containing the maximum number of cycles is $T_k(n)$. This resolves both a…

We characterize the connected graphs of given order $n$ and given independence number $\alpha$ that maximize the number of maximum independent sets. For $3\leq \alpha\leq n/2$, there is a unique such graph that arises from the disjoint…

A well-known result of Ajtai et al. from 1982 states that every $k$-graph $H$ on $n$ vertices, with girth at least five, and average degree $t^{k-1}$ contains an independent set of size $c n (\log t)^{1/(k-1)}/t$ for some $c>0$. In this…

We study the function $H_n(C_{2k})$, the maximum number of Hamilton paths such that the union of any pair of them contains $C_{2k}$ as a subgraph. We give upper bounds on this quantity for $k\ge 3$, improving results of Harcos and…

The generalized Petersen graph $G(n, k)$ is a cubic graph with vertex set $V(G(n, k)) = \{v_i\}_{0 \leq i < n} \cup \{w_i\}_{0 \leq i < n}$ and edge set $E(G(n, k)) = \{v_i v_{i+1}\}_{0 \leq i < n} \cup \{w_i w_{i+k}\}_{0 \leq i < n} \cup…

For integers $k,n$ with $1 \le k \le n/2$, let $f(k,n)$ be the smallest integer $t$ such that every $t$-connected $n$-vertex graph has a spanning bipartite $k$-connected subgraph. A conjecture of Thomassen asserts that $f(k,n)$ is upper…

For all integers $k$ with $k\geq 2$, if $G$ is a balanced $k$-partite graph on $n\geq 3$ vertices with minimum degree at least \[…