Kneser hypergraphs Frédéric Meunier May 21th, 2015 CERMICS, Optimisation et Systèmes
Kneser hypergraphs m, l, r three integers s.t. m rl. Kneser hypergraph KG r (m, l): V (KG r (m, l)) = ( [m]) l { E(KG r (m, l)) = {A 1,..., A r } : A i ( [m]) l, Ai A j = for i j}
Chromatic number Theorem (Alon-Frankl-Lovász theorem) χ(kg r m r(l 1) (m, l)) = r 1 All proofs: if true for r 1 and r 2, then true for r 1 r 2. true when r is prime. Original proof for the case r prime: as for Lovász-Kneser conjecture, uses box complexes but Dold s theorem instead BU theorem.
A combinatorial proof Ziegler (2003) proposed a combinatorial proof via a Z p -Tucker s lemma. Assume p prime and KG p (m, l) properly colored with t colors. Z p = pth roots of unity With the help of coloring, build a map λ : (Z p {0}) m \ {0} Z p [t + pl 2] x (s(x), v(x) ) }{{}}{{} sign absolute value satisfying condition of a Z p-tucker lemma λ(ωx) = ωλ(x) for ω Z p condition on {λ(x 1 ),..., λ(x p )} when x 1 x p. Second point satisfied by coloring condition: no p adjacent vertices get the same color. Thus, (p 1)(t 1) + pl 1 m, i.e. t m p(l 1) p 1
Z p -Tucker lemma Lemma. Let α {0, 1,..., k}. If there exists λ : (Z p {0}) m \ {0} Z p [k] x (s(x), v(x)) such that λ(ωx) = ωλ(x) for ω Z p v(x) = v(y) α and x y = s(x) = s(y) v(x 1 ) = = v(x p ) α + 1 and x 1 x p = s(x i ) s not pairwise distinct, then m α + (k α)(p 1). Octahedral Tucker s lemma. Let λ : {+,, 0} m \ {0} {±1,..., ±k} s.t. λ( x) = λ(x) x y = λ(x) + λ(y) 0 Then m k.
Z p = {1, g, g 2,..., g p 1 }. 1 g g 2 g 3 g 4 1 g g 2 g 3 g 4 λ α (g 2, g, 0, g 3, g 3, g 2, g 2, 0, g 4 )
K = Z p Z p }{{} m times L = Z p Z p } {{ } α times Z p -Tucker: a proof p 1 p 1 }{{} k α times Lemma (Z p -Tucker lemma reformulation) If there exists λ : sd(k) Zp L, then m α + (k α)(p 1). Proof. Dold s theorem: connectivity(sd(k)) < dim(l). connectivity(sd(k)) = m 2 and dim(l) = α + (k α)(p 1) 1.
Ziegler s proof c : ( ) [m] l [t] proper coloring of KG p (m, l) with t colors. Extension for any U [m]: c(u) = max{c(a) : A U, A = l}. x ω = {i : x i = ω} λ(x) = (x a, x ) if x p(l 1) and a = min{i : x i 0} (ω, c(x ω ) + p(l 1)) if x p(l 1) + 1 and ω = arg max{c(x ω ) : ω Z p} α = p(l 1).
A Zig-zag theorem for Kneser hypergraphs Theorem (M. 2014) Let p be a prime number. Any proper coloring c of KG p (m, l) with t colors contains a complete p-uniform p-partite hypergraph with parts U 1,..., U p satisfying the following properties. It has m p(l 1) vertices. The values of U j differ by at most one. The vertices of U j get distinct colors. Generalizes Alon-Frankl-Lovász theorem and almost generalizes the Zig-zag theorem for Kneser graphs. Proof uses a Z p -Fan lemma due to Hanke, Sanyal, Schultz, Ziegler (2009), M. (2006)
Z p -Fan lemma Theorem Let be such that λ : (Z p {0}) m \ {0} Z p [k] x (s(x), v(x)) λ(ωx) = ωλ(x) for ω Z p v(x) = v(y) and x y = s(x) = s(y). Then there exists an m-chain x 1 x m such that λ({x 1,..., x m }) = {(s 1, v 1 ),..., (s m, v m )} with v 1 < < v m and s i s i+1 for i [m 1].
Local chromatic number of Kneser hypergraphs Let H = (V, E) be a uniform hypergraph. For X V, N [X] := X N (X). N (X) = {v : e E s.t. e \ X = {v}}. ψ(h) = min c max e E, v e c(n [e \ {v}]), where minimum taken over all proper colorings c. Consequence of the Zig-zag theorem for Kneser hypergraphs: Theorem ( ) ψ(kg p m p(l 1) m p(l 1) (m, l)) min + 1, p p 1 for any prime number p.
Hedetniemi s conjecture for Kneser hypergraphs Theorem (Hajiabolhassan-M. 2014) χ ( KG r (m, l) KG r (m, l ) ) = min ( χ ( KG r (m, l) ), χ ( KG r (m, l ) ))
Product of hypergraphs and Zhu s conjecture u 1 v 1 {u 1, u 2, u 3} E(G) {v 1, v 2, v 3, v 4} E(H) u 2 v 2 {(u 1, v 1), (u 1, v 2), (u 2, v 1), (u 3, v 3), (u 3, v 4)} E(G H) u 3 v 3 v 4 Let G, H be two hypergraphs. Categorical product G H: V (G H) = V (G) V (H) E(G H) = {e V (G H) : π G (e) E(G), π H (e) E(H)} Proper coloring of a hypergraph: no monochromatic edges. Zhu s conjecture (1992) χ(g H) = min(χ(g), χ(h))
Product of graphs vs of hypergraphs, and coloring Product of two graphs seen as hypergraphs: will have edges of cardinality 3 and 4. G H G H G H G H G H G H But the chromatic number does not depend of the definition of the product. G H Zhu s conjecture is a true generalization of Hedetniemi s conjecture.
New graphs for Hedetniemi s conjecture U 1,..., U s pairwise disjoint, l 1,..., l s integers B = { A ( ) } [n] : A U i l i k bases of the truncation of a partition matroid Proposition (Hajiabolhassan-M.) Graphs KG 2 (B) satisfy Hedetniemi s conjecture. Holds also if B is the set of spanning trees of some dense graph.
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