C是G中的有向圈所形成的集合，其中C是G的有向圈雙重覆蓋，我們將
C當成頂點集合，記為I_C，如果|C∩C'|=k則我們稱C和C'這兩個有向圈為k-連通。
 令ϕ^*_c(G)=min{χ_c(I_C)}，其中ϕ_c^*(G)是所有有向圈雙重覆蓋著色數中最小的那個，非常容易證明對於所有的圖G，ϕ_c(G)<ϕ_c^*(G)。
    同時我們也在此說明存在圖G使得ϕ_c(G)<ϕ_c^*(G)。
    令J_(2k+1)是一個史納克圖形，頂點集合V(J_(2k+1) )={a_i,b_i,c_i,d_i:i=0,1,…,2k}
邊集合E(J_(2k+1) )={b_i a_i,b_i c_i,b_i d_i,a_i a_(i+1),c_i d_(i+1),d_i c_(i+1):i=0,1,…,2k}，其中每一個索引要模2k+1。
    在本篇論文中，我們證明了：
(1)If k≥1,then χ_c(I_C )≥5.

For a set C of directed crcuits of a graph G that form an oriented circuit double cover,we denote by I_C the graph with vertex set C,in which two circuits C and C' are connected by k edges if |C∩C'|=k.
    Let ϕ^*_c (G)=min{χ_c(I_C)},where the minimum is taken over all the oriented circuit double covers of G,it is easy to show that for any graph G,ϕ_c(G)<ϕ^*_c (G).
    We also show that there are graphs G for which ϕ_c (G)<ϕ^*_c (G).
    Let J_(2k+1) be the flower snark which has vertex set V(J_(2k+1) )={a_i,b_i,c_i,d_i:i=0,1,…,2k}
 and dege set E(J_(2k+1) )={b_i a_i,b_i c_i,b_i d_i,a_i a_(i+1),c_i d_(i+1),d_i c_(i+1):i=0,1,…,2k},where the summationin indices are modulo 2k+1.
    In this thesis,we proved that
(1)If k≥1,then 〖 χ〗_c (I_C )≥5.

目  錄

第一章：簡介..........................................1

第二章：主定理證明.....................................6

  2.1：基本定義介紹及圈的性質...........................6

  2.2：各類大圈的證明.................................14

參考文獻.............................................40

圖目錄

圖1  圖..................................................1
圖2  圖1的有向圖........................................1
圖3  P_4..................................................1
圖4  C_5(右:有向圈).......................................1
圖5  雙重圈覆蓋C={C_1,C_2,C_3,C_4 }..........................2
圖6  G_1:平面圖；G_2:非平面圖..............................2
圖7  左:χ_C(G)=5/2  右:Φ_C(G^* )=5/2..................2
圖8  J_9...................................................5
圖9  Flower snark(J_9)....................................6
圖10 Flower snark圖中的Y處..............................6
圖11 Flower snark圖中的H處..............................7
圖12  虛線:左、右(S型)皆為大圈    實線:小圈..................7
圖13 e無法被通過.........................................9
圖14 小圈不能同時在Y處回頭..............................10
圖15 C_1相交每個H_i兩次...................................16
圖16 C_1走S形，長度為1...................................16
圖17 S形內部小圈長度為六.................................16
圖18 S形兩側Y處.........................................17
圖19 小圈相交情況.......................................19
圖20 小圈相交大圈的情況.................................19
圖21 Y處有相異大圈相交..................................20
圖22 史納克圖形的轉換...................................22
圖23 圖22的圈相交圖....................................22
圖24 四個大圈相交情形...................................23
圖25 圈的交集圖.........................................23
圖26 兩個Y處有大圈相交..................................25
圖27 四個大圈中存在S形.................................26
圖28 圖27的圈相交圖....................................26
圖29 圖27的轉換........................................27
圖30 大圈一定要走Y處....................................28
圖31 兩個大圈和Y不相交..................................29
圖32 兩個大圈在Y處不對稱...............................30
圖33 兩大圈連續不相交...................................31
圖34 H處的奇、偶性質....................................31
圖35 S形內部............................................32
圖36 大圈相交的交集圖長相...............................33
圖37 S形的圈交集圖......................................33
圖38 圈交集圖間的相連...................................34
圖39 圈交集圖對應關係...................................34
圖40 圈交集圖對應關係...................................34
圖41 圈交集圖對應關係...................................34
圖42 與S形的對應關係...................................35
圖43 與S形的對應關係...................................35
圖44 S形與S形之間的對應關係............................35
圖45 S形之間對應關係....................................35
圖46 S形之間對應關係....................................35
圖47 兩個大圈...........................................37
圖49 圈的交集圖.........................................38

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