Hamiltonian cycles in neighbor-swap graphs : master's thesis

Smerdu, Ana (author), Verhoeff, Tom (mentor), Žitnik, Arjana (co-mentor)

In 1965 Derrick Henry Lehmer conjectured that every neighbor-swap graph admits an imperfect Hamiltonian path. This path, also known as Lehmer path, is a walk visiting all the vertices of a graph where some of them might be visited twice in a row. For most of the neighbor-swap graphs the conjecture is already proved, it remains open only for two families of graphs. We will present known results with their proofs in the thesis. It turns out most of these graphs even contain a Lehmer cycle and we will show how to construct them. For the missing part of the proof we will present a possible approach, that might finally confirm D. H. Lehmer's conjecture. First we find a Hamiltonian path in a related binary neighbor-swap graph and then step by step add the missing symbols, connecting the paths together into a Lehmer path.

• Language: English
• Keywords: neighbor-swap graphs, multisets, permutations, transpositions, Hamiltonian paths, Lehmer paths, posets
• Work type: Master's thesis/paper
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2019
• PID: 20.500.12556/RUL-108786
• UDC: 519.17
• COBISS.SI-ID: 18676313
• Publication date in RUL: 25.07.2019

Secondary language

• Language: Slovenian
• Title: Hamiltonovi cikli v grafih transpozicij sosedov

Abstract

Lehmerjeva pot je nepopolna Hamiltonova pot in je definirana kot sprehod, kjer obiščemo vsa vozlišča v grafu, nekatera pa lahko obiščemo dvakrat zapored. Leta 1965 je Derrick Henry Lehmer postavil domnevo, da vsak graf transpozicij sosedov vsebuje Lehmerjevo pot. Domneva je v veliki meri že dokazana in te izreke z dokazi bomo v magistrski nalogi tudi predstavili. Izkaže se, da je v mnogih primerih možna celo konstrukcija Lehmerjevega cikla, kar bomo tudi pokazali. Za mankajoči del dokaza bomo podali možen pristop, ki bi domnevo D. H. Lehmerja dokončno potrdil. Najprej poiščemo Hamiltonovo pot v sorodnem dvojiškem grafu transpozicij sosedov, nato pa s konstrukcijo Stachowiaka postopoma dodajamo manjkajoče simbole in združujemo dobljene Lehmerjeve poti.

• Keywords: graf transpozicij sosedov, večkratne množice, permutacije, transpozicije, Hamiltonove poti, Lehmerjeve poti, delno urejene množice