Kneser's conjecture, chromatic number, and homotopy

AbstractIf the simplicial complex formed by the neighborhoods of points of a graph is (k − 2)-connected then the graph is not k-colorable. As a corollary Kneser's conjecture is proved, asserting that if all n-subsets of a (2n − k)-element set are divided into k + 1 classes, one of the classes contains two disjoint n-subsets

The multichromatic numbers of some Kneser graphs

AbstractThe Kneser graph K(m,n) has the n-subsets of {1,2,…,m} as its vertices, two such vertices being adjacent whenever they are disjoint. The kth multichromatic number of the graph G is the least integer t such that the vertices of G can be assigned k-subsets of {1,2, …, t}, so that adjacent vertices of G receive disjoint sets. The values of Xk(K(m,n)) are computed for n = 2, 3 and bounded for n ⩾ 4

Homology of Hom Complexes

The h The hom complex Hom(G,K) is the order complex of the poset composed of the graph multihomomorphisms from G to K. We use homology to provide conditions under which the hom complex is not contractible and derive a lower bound on the rank of its homology groups.om complex Hom(G,K) is the order complex of the poset composed of the graph multihomomorphisms from G to K. We use homology to provide conditions under which the hom complex is not contractible and derive a lower bound on the rank of its homology groups

A generalization of Gale's lemma

In this work, we present a generalization of Gale's lemma. Using this generalization, we introduce two combinatorial sharp lower bounds for ${\rm conid}({\rm B}_0(G))+1$ and ${\rm conid}({\rm B}(G))+2$, two famous topological lower bounds for the chromatic number of a graph $G$

A class of additive multiplicative graph functions

AbstractFor a fixed graph G, the capacity function for G, PG, is defined by PG(H) = limn→∞[γG(Hn)]1/n, where γG(H) is the maximum number of disjoint G's in H. In [2], Hsu proved that PK2 is multiplicative or not. In this paper, we prove that PG is multiplicative and additive for some graphs G which include K2. Some properties of PG are also discussed in this paper

On endo-homology of complexes of graphs

AbstractLet L be a subcomplex of a complex K. If the homomorphism from inclusion i∗:Hq(L)→Hq(K) is an isomorphism for all q ⩾ 0, then we say that L and K are endo-homologous. The clique complex of a graph G, denoted by C(G), is an abstract complex whose simplices are the cliques of G. The present paper is a generalization of Ivashchenko (1994) along several directions. For a graph G and a given subgraph F of G, some necessary and sufficient conditions for C(G) to be endo-homologous to C(F) are given. Similar theorems hold also for the independence complex I(G) of G, where I(G) − C(Gc), the clique complex of the complement of G