Sparse Kneser graphs are Hamiltonian

Bipartite Kneser Graphs are Hamiltonian

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 (=sets) that are disjoint. The bipartite Kneser graph H(n, k) has as vertices all k-element and (n−k)-element subsets of [n] and an edge between any two vertices where one is a subset of the other. It has long been conjectured that all connected Kneser graphs and bipar...

KNESER GRAPHS ARE HAMILTONIAN FOR n

Hamiltonian Kneser Graphs

The Kneser graph K (n; k) has as vertices the k-subsets of f1;2;:::;ng. Two vertices are adjacent if the corresponding k-subsets are disjoint. It was recently proved by the rst author 2] that Kneser graphs have Hamilton cycles for n 3k. In this note, we give a short proof for the case when k divides n. x 1. Preliminaries. Suppose that n k 1 are integers and let n] := f1; 2; :::; ng. We denote t...

An Inductive Construction for Hamilton Cycles in Kneser Graphs

The Kneser graph K(n, r) has as vertices all r-subsets of an n-set with two vertices adjacent if the corresponding subsets are disjoint. It is conjectured that, except for K(5, 2), these graphs are Hamiltonian for all n ≥ 2r +1. In this note we describe an inductive construction which relates Hamiltonicity of K(2r + 2s, r) to Hamiltonicity of K(2r′+s, r′). This shows (among other things) that H...

Binomial and Q-Binomial Coefficient Inequalities Related to the Hamiltonicity of the Kneser Graphs and Their Q-Analogues

The Kneser graph K(n, k) has as vertices all the k-subsets of a fixed n-set and has as edges the pairs [A, B] of vertices such that A and B are disjoint. It is known that these graphs are Hamiltonian if ( n&1 k&1) ( n&k k ) for n 2k+1. We determine asymptotically for fixed k the minimum value n=e(k) for which this inequality holds. In addition we give an asymptotic formula for the solution of k...