Results 1  10
of
31
Complexes of graph homomorphisms
 Israel J. Math
"... Abstract. Hom (G, H) is a polyhedral complex defined for any two undirected graphs G and H. This construction was introduced by Lovász to give lower bounds for chromatic numbers of graphs. In this paper we initiate the study of the topological properties of this class of complexes. We prove that Hom ..."
Abstract

Cited by 48 (10 self)
 Add to MetaCart
(Show Context)
Abstract. Hom (G, H) is a polyhedral complex defined for any two undirected graphs G and H. This construction was introduced by Lovász to give lower bounds for chromatic numbers of graphs. In this paper we initiate the study of the topological properties of this class of complexes. We prove that Hom (Km, Kn) is homotopy equivalent to a wedge of (n−m)dimensional spheres, and provide an enumeration formula for the number of the spheres. As a corollary we prove that if for some graph G, and integers m ≥ 2 and k ≥ −1, we have ̟k 1 (Hom (Km, G)) ̸ = 0, then χ(G) ≥ k + m; here Z2action is induced by the swapping of two vertices in Km, and ̟1 is the first StiefelWhitney class corresponding to this action. Furthermore, we prove that a fold in the first argument of Hom (G, H) induces a homotopy equivalence. It then follows that Hom (F, Kn) is homotopy equivalent to a direct product of (n−2)dimensional spheres, whileHom (F, Kn) is homotopy equivalent to a wedge of spheres, where F is an arbitrary forest and F is its complement. 1.1. Definition of the main object. 1.
Local chromatic number, Ky Fan’s theorem, and circular colorings
 Combinatorica
"... The local chromatic number of a graph was introduced in [12]. It is in between the chromatic and fractional chromatic numbers. This motivates the study of the local chromatic number of graphs for which these quantities are far apart. Such graphs include Kneser graphs, their vertex colorcritical sub ..."
Abstract

Cited by 27 (6 self)
 Add to MetaCart
(Show Context)
The local chromatic number of a graph was introduced in [12]. It is in between the chromatic and fractional chromatic numbers. This motivates the study of the local chromatic number of graphs for which these quantities are far apart. Such graphs include Kneser graphs, their vertex colorcritical subgraphs, the Schrijver (or stable Kneser) graphs; Mycielski graphs, and their generalizations; and Borsuk graphs. We give more or less tight bounds for the local chromatic number of many of these graphs. We use an old topological result of Ky Fan [14] which generalizes the BorsukUlam theorem. It implies the existence of a multicolored copy of the complete bipartite graph K⌈t/2⌉,⌊t/2 ⌋ in every proper coloring of many graphs whose chromatic number t is determined via a topological argument. (This was in particular noted for Kneser graphs by Ky Fan [15].) This yields a lower bound of ⌈t/2 ⌉ + 1 for the local chromatic number of these graphs. We show this bound to be tight or almost tight in many cases. As another consequence of the above we prove that the graphs considered here have equal circular and ordinary chromatic numbers if the latter is even. This partially proves a conjecture of Johnson, Holroyd, and Stahl and was independently attained by F. Meunier
Proof of the Lovász conjecture
 ANN. OF MATH
, 2008
"... In this paper we prove the Lovász Conjecture: If Hom(C2r+1, H) is kconnected, then χ(H) ≥ k + 4, where H is a finite undirected graph, C2r+1 is a cycle with 2r+1 vertices, r, k ∈ Z, r ≥ 1, k ≥ −1, and Hom(G, H) is the cell complex with the vertex set being the set of all graph homomorphisms from ..."
Abstract

Cited by 23 (1 self)
 Add to MetaCart
In this paper we prove the Lovász Conjecture: If Hom(C2r+1, H) is kconnected, then χ(H) ≥ k + 4, where H is a finite undirected graph, C2r+1 is a cycle with 2r+1 vertices, r, k ∈ Z, r ≥ 1, k ≥ −1, and Hom(G, H) is the cell complex with the vertex set being the set of all graph homomorphisms from G to H, and cells all allowed list Hcolorings of G. Our method is to compute, by means of spectral sequences, the obstructions to graph colorings, which lie either directly in the cohomology groups of Hom(C2r+1, Kn), or in the vanishing of the certain powers of StiefelWhitney classes of Hom(C2r+1, Kn), viewed as Z2spaces, resulting in proving even sharper statements.
Topological Lower Bounds for the Chromatic Number: A Hierarchy
 JAHRESBER. DEUTSCH. MATH.VEREIN
, 2004
"... Kneser's conjecture, first proved by Lovasz in 1978, states that the graph with all k element subsets of 2, . . . , n} as vertices and all pairs of disjoint sets as edges has chromatic number n2k+ 2. Several other proofs have been published (by Barany, Schrijver, Dol'nikov, Sarkari ..."
Abstract

Cited by 14 (1 self)
 Add to MetaCart
Kneser's conjecture, first proved by Lovasz in 1978, states that the graph with all k element subsets of 2, . . . , n} as vertices and all pairs of disjoint sets as edges has chromatic number n2k+ 2. Several other proofs have been published (by Barany, Schrijver, Dol'nikov, Sarkaria, Krz, Greene, and others), all of them based on the BorsukUlam theorem from algebraic topology, but otherwise quite di#erent. Each can be extended to yield some lower bound on the chromatic number of an arbitrary graph. We show
Topological obstructions to graph colorings
 ELECTRON. RES. ANNOUNC. AMER. MATH. SOC
, 2003
"... For any two graphs G and H Lovász has defined a cell complex Hom(G, H) having in mind the general program that the algebraic invariants of these complexes should provide obstructions to graph colorings. Here we announce the proof of a conjecture of Lovász concerning these complexes with G a cycle ..."
Abstract

Cited by 6 (1 self)
 Add to MetaCart
For any two graphs G and H Lovász has defined a cell complex Hom(G, H) having in mind the general program that the algebraic invariants of these complexes should provide obstructions to graph colorings. Here we announce the proof of a conjecture of Lovász concerning these complexes with G a cycle of odd length. More specifically, we show that If Hom(C2r+1, G) is kconnected, then χ(G)≥k + 4. Our actual statement is somewhat sharper, as we find obstructions already in the nonvanishing of powers of certain StiefelWhitney classes.
On the chromatic number of Kneser hypergraphs
, 2000
"... We give a simple and elementary proof of Kr'iz's lower bound on the chromatic number of the Kneser rhypergraph of a set system S. ..."
Abstract

Cited by 5 (1 self)
 Add to MetaCart
We give a simple and elementary proof of Kr'iz's lower bound on the chromatic number of the Kneser rhypergraph of a set system S.
Combinatorial Stokes formulas via minimal resolutions
, 2007
"... We describe an explicit chain map from the standard resolution to the minimal resolution for the finite cyclic group Zk of order k. We then demonstrate how such a chain map induces a “Zkcombinatorial Stokes theorem”, which in turn implies “Dold’s theorem ” that there is no equivariant map from an n ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
We describe an explicit chain map from the standard resolution to the minimal resolution for the finite cyclic group Zk of order k. We then demonstrate how such a chain map induces a “Zkcombinatorial Stokes theorem”, which in turn implies “Dold’s theorem ” that there is no equivariant map from an nconnected to an ndimensional free Zkcomplex. Thus we build a combinatorial access road to problems in combinatorics and discrete geometry that have previously been treated with methods from equivariant topology. The special case k = 2 for this is classical; it involves Tucker’s (1949) combinatorial lemma which implies the Borsuk–Ulam theorem, its proof via chain complexes by Lefschetz (1949), the combinatorial Stokes formula of Fan (1967), and Meunier’s work (2006).
Stable Kneser hypergraphs and ideals in N with the Nikodým property
 Proceedings of the American Mathematical Society 137
, 2009
"... ..."
(Show Context)
THE BORSUKULAMPROPERTY, TUCKERPROPERTY AND CONSTRUCTIVE PROOFS IN COMBINATORICS
, 2005
"... This article is concerned with a general scheme on how to obtain constructive proofs for combinatorial theorems that have topological proofs so far. To this end the combinatorial concept of Tuckerproperty of a finite group G is introduced and its relation to the topological BorsukUlamproperty is ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
This article is concerned with a general scheme on how to obtain constructive proofs for combinatorial theorems that have topological proofs so far. To this end the combinatorial concept of Tuckerproperty of a finite group G is introduced and its relation to the topological BorsukUlamproperty is discussed. Applications of the Tuckerproperty in combinatorics are demonstrated.