Related papers: Paths of homomorphisms from stable Kneser graphs

In this article, we prove that the neighborhood complex of the Kneser graph $KG_{3,k}$ is of the same homotopy type as that of a wedge of $\frac{(k+1)(k+3)(k+4)(k+6)}{4}+1$ spheres of dimension $k$. We construct a maximal subgraph $S_{3,k}$…

The Kneser graph $K(n,k)$ is defined for integers $n$ and $k$ with $n \geq 2k$ as the graph whose vertices are all the $k$-subsets of $\{1,2,\ldots,n\}$ where two such sets are adjacent if they are disjoint. A classical result of Lov\'asz…

For given graphs $G$ and $H$, let $|Hom(G,H)|$ denote the set of graph homomorphisms from $G$ to $H$. We show that for any finite, $n$-regular, bipartite graph $G$ and any finite graph $H$ (perhaps with loops), $|Hom(G,H)|$ is maximum when…

Assume $G$ is a graph. We view $G$ as a symmetric digraph, in which each edge $uv$ of $G$ is replaced by a pair of opposite arcs $e=(u,v)$ and $e^{-1}=(v,u)$. Assume $S$ is an inverse closed subset of permutations of positive integers. We…

\noindent In this paper, we show that for any positive integers $r$, $k$, $\Theta$, and $\Gamma$ such that $k \geq 2$ and $r \geq k + \Gamma$, there exists a connected graph $G$ for which $$\begin{array}{llcr} \omega (G) = \chi (G) = k, &…

A $k$-uniform hypergraph $M$ is set-homogeneous if it is countable (possibly finite) and whenever two finite induced subhypergraphs $U,V$ are isomorphic there is $g\in Aut(M)$ with $U^g=V$; the hypergraph $M$ is said to be homogeneous if in…

A finite graph $\G$ is said to be {\em $(G,3)$-$($connected$)$ homogeneous} if every isomorphism between any two isomorphic (connected) subgraphs of order at most $3$ extends to an automorphism $g\in G$ of the graph, where $G$ is a group of…

The stability number alpha(G) of a graph G is the cardinality of a maximum stable set in G, xi(G) denotes the size of core(G), where core(G) is the intersection of all maximum stable sets of G. In this paper we prove that for a graph G…

The Hom complex ${\rm Hom}(T,G)$ of graphs is a CW-complex associated to a pair of graphs $T$ and $G$, considered in the graph coloring problem. It is known that certain homotopy invariants of ${\rm Hom}(T,G)$ give lower bounds for the…

In the way of proving Kneser's conjecture, L\'{a}szl\'{o} Lov\'{a}sz settled out a new lower bound for the chromatic number of graphs. He showed that if the hom complex $||Hom(\mathcal{K}_2, H)||$ of a graph $H$ is topologically…

We show P\'eter Csorba's conjecture that the graph homomorphism complex Hom(C_5,K_{n+2}) is homeomorphic to a Stiefel manifold, the space of unit tangent vectors to the n-dimensional sphere. For this a general tool is developed that allows…

The Johnson graph $J(n, i)$ is defined as the graph whose vertex set is the set of all $i$-element subsets of $\{1, . . ., n \}$, and two vertices are adjacent whenever the cardinality of their intersection is equal to $i$-1. In Ramras and…

Given a natural number k and an orientable surface S of finite type, define the k-curve graph to be the graph with vertices corresponding to isotopy classes of essential simple closed curves on S and with edges corresponding to pairs of…

We classify all the $2$-arc-transitive strongly regular graphs, and use this classification to study the family of finite $(G,3)$-geodesic-transitive graphs of girth $4$ or $5$ for some group $G$ of automorphisms. For this application we…

The Kneser graph $K(n, k)$ has as vertices all $k$-element subsets of $[n]=\{1,2,...,n \}$ and an edge between any two vertices that are disjoint. If $n=2k+1$, then $K(n, k)$ is called an odd graph. Let $ n >4$ and $1< k < \frac{n}{2} $. In…

The Hom complex $\mathrm{Hom}(G, H)$ of graphs is a simplicial complex associated to a pair of graphs $G$ and $H$, and its homotopy type is of interest in the graph coloring problem and the homomorphism reconfiguration problem. In this…

A geometric graph \G is a simple graph drawn in the plane, on points in general position, with straight-line edges. We call \G a geometric realization of the underlying abstract graph G. A geometric homomorphism from \G to \H is a vertex…

Let $V(n,k,s)$ be the set of $k$-subsets $S$ of $[n]$ such that for all $i,j\in S$, we have $|i-j|\geq s$ We define almost $s$-stable Kneser hypergraph $KG^r{{[n]}\choose k}_{s{\tiny{\textup{-stab}}}}^{\displaystyle\sim}$ to be the…

A set $\mathcal{F} \subset \operatorname{Sym}(V)$ is \textsl{intersecting} if any two of its elements agree on some element of $V$. Given a finite transitive permutation group $G\leq \operatorname{Sym}(V)$, the \textsl{intersection density}…

To any two graphs G and H one can associate a cell complex Hom(G,H) by taking all graph multihomorphisms from G to H as cells. In this paper we prove the Lovasz Conjecture which states that if Hom(C_{2r+1},G) is k-connected, then…