| Abstract: | 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. |
