Bipartite Kneser Graphs are Hamiltonian

KNESER GRAPHS ARE HAMILTONIAN FOR n

Sparse Kneser graphs are Hamiltonian

For integers k ≥ 1 and n ≥ 2k + 1, the Kneser graph K(n, k) is the graph whose vertices are the k-element subsets of {1, . . . , n} and whose edges connect pairs of subsets that are disjoint. The Kneser graphs of the form K(2k + 1, k) are also known as the odd graphs. We settle an old problem due to Meredith, Lloyd, and Biggs from the 1970s, proving that for every k ≥ 3, the odd graph K(2k + 1,...

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...

Colorful subgraphs in Kneser-like graphs

Combining Ky Fan’s theorem with ideas of Greene and Matoušek we prove a generalization of Dol’nikov’s theorem. Using another variant of the Borsuk-Ulam theorem due to Bacon and Tucker, we also prove the presence of all possible completely multicolored t-vertex complete bipartite graphs in t-colored t-chromatic Kneser graphs and in several of their relatives. In particular, this implies a genera...