Hamiltonian paths in odd graphs

Letícia Bueno,Figueiredo De,Luerbio Faria,Fonseca Da

doi:10.2298/aadm0902386b

Letícia Bueno, Figueiredo De + Show 2 more

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https://doi.org/10.2298/aadm0902386b

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Abstract

Lov?sz conjectured that every connected vertex-transitive graph has a Hamiltonian path. The odd graphs Ok form a well-studied family of connected, k-regular, vertex-transitive graphs. It was previously known that Ok has Hamiltonian paths for k ? 14. A direct computation of Hamiltonian paths in Ok is not feasible for large values of k, because Ok has (2k - 1, k - 1) vertices and k/2 (2k - 1, k - 1) edges. We show that Ok has Hamiltonian paths for 15 ? k ? 18. Instead of directly running any heuristics, we use existing results on the middle levels problem, therefore further relating these two fundamental problems, namely finding a Hamiltonian path in the odd graph and finding a Hamiltonian cycle in the corresponding middle levels graph. We show that further improved results for the middle levels problem can be used to find Hamiltonian paths in Ok for larger values of k.

Highlights

A spanning cycle in a graph is a Hamiltonian cycle and a graph which contains such cycle is said to be Hamiltonian
Odd graphs, Kneser graphs, middle levels problem. 1A preliminary version of this work was presented at SGT in Rio – Workshop on Spectral Graph Theory with applications on Computer Science, Combinatorial optimization and Chemistry
Rio de Janeiro 2008. 2Research supported by CNPq grant. 3Research supported by CNPq grant PDJ-151194/2007-6

Summary

INTRODUCTION

A spanning cycle in a graph is a Hamiltonian cycle and a graph which contains such cycle is said to be Hamiltonian. Lovasz [11] conjectured that every connected vertex-transitive graph has a Hamiltonian path. Odd graphs, Kneser graphs, middle levels problem. Shields and Savage [18] recently showed that Bk has Hamiltonian cycles for k = 17 and 18 These results [5, 6, 16, 18] were based on properties of smaller graphs obtained by identifying vertices that are equivalent according to some equivalence relation. Savage and Winkler [15] showed that if Bk has a Hamiltonian cycle for k ≤ h, Bk has a cycle containing a fraction 1 − ε of the graph vertices for all k, where ε is a function of h.

THE REDUCED GRAPH

HAMILTONIAN PATHS IN THE ODD GRAPH

Findings

CONCLUSION AND OPEN PROBLEMS