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    Mutually Independent Hamiltonian Cycles on Cartesian Product Graphs

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    [[abstract]]在一個圖形(graph) G 中,一個迴圈(cycle) C = ?v0, v1, ..., vk, v0?,定義 為一個相鄰(adjacent) 的點所組成的序列,且對所有1 ? i < j ? k,vi ≠ vj。 而在G 中,一個迴圈若包含了圖形G 中所有的點,則稱此迴圈為漢米爾頓迴 圈(Hamiltonian cycle)。若圖形G 中存在漢米爾頓迴圈則稱此圖為漢米爾頓 圖(Hamiltonian graph)。G 中的兩個漢米爾頓迴圈C1 = ?u0, u1, u2, ..., un?1, u0? 及C2 = ?v0, v1, v2, ..., vn?1, v0? 稱為獨立(independent) 則表示u0 = v0 且對所有 1 ? i ? n?1,ui ≠ vi。若稱G 中的一個漢米爾頓迴圈集合C = {C1,C2, ...,Ck} 為互相獨立(imutually independent) 的意思是,集合中任兩個漢米爾頓迴圈 皆為獨立。一個圖形G 中的互相獨立漢米爾頓數(imutually independent Hamiltonianicity) 以IHC(G) = k 表示,其定義為max{k ? N| 從圖形中任意 一點u 作為起點,皆存在k 個互相獨立漢米爾頓迴圈}。兩圖形G 與H 的 卡式積(Cartesian product) 為一個新的圖形並以G × H 表示,其點集合為 V (G) × V (H),此圖形中若兩點(u, v) 與(u′, v′) 相連必須滿足:(1) u = u′ 且 vv′ ? E(H),或(2) v = v′ 且uu′ ? E(G)。 在本論文中,我們針對了圖形G = G1 × G2,討論其互相獨立漢米爾頓 數的性質,其中G1 及G2 為漢米爾頓圖。我們證明了在給定不同的條件下, IHC(G1 × G2) ? IHC(G1) 或IHC(G1) + 2。此外,我們參考環方格(toroidal mesh) 圖形的定義,其等價於兩個迴圈做卡式積的圖形Cm × Cn,其中m, n 分 別代表兩迴圈的長度。本論文中也證明了對任意兩個正整數m, n ? 3 而言, IHC(Cm × Cn) = 4。[[abstract]]In a graph G, a cycle C = ?v0, v1, ..., vk, v0? is defined as a sequence of adjacent vertices and for all 0 ? i < j ? k, vi ≠ vj ; a cycle is called Hamiltonian cycle if it contains all vertices of G. If there exists a Hamiltonian cycle in G, then G is a Hamiltonian graph. Two Hamiltonian cycles C1 = ?u0, u1, u2, ..., un?1, u0? and C2 = ?v0, v1, v2, ..., vn?1, v0? are independent if u0 = v0, ui ≠ vi for all 1 ? i ? n ? 1. A set of Hamiltonian cycles C = {C1,C2, ...,Ck} of G are mutually independent if any two different Hamiltonian cycles of C are independent. The mutually independent Hamiltonianicity of graph G, namely IHC(G) = k, is the maximum integer k such that for any vertex u of G there exists k-mutually independent Hamiltonian cycles starting at u. The Cartesian product of graphs G and H, written by G×H, is the graph with vertex set V (G)×V (H) specified by putting (u, v) adjacent to (u′, v′) if and only if (1) u = u′ and vv′ ? E(H), or (2) v = v′ and uu′ ? E(G). In this thesis, we study mutually independent Hamiltonianicity on G = G1×G2, where G1 and G2 are Hamiltonian graphs. We prove that IHC(G1×G2) ? IHC(G1) or IHC(G1) + 2 when given some difference conditions. We refer to toroidal mesh graph and define graph Cm × Cn, where m, n are lengths of cycles. We show that IHC(Cm × Cn) = 4 for any positive integers m, n ? 3.[[note]]碩
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