基本介紹
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圖論中有兩類相互有一定聯繫然而又是性質不同的問題,一類是研究用某種類型的子圖去覆蓋圖G的點集或邊集(即,或),這就是所謂覆蓋問題。例如正常邊染色是用邊不交的邊無關集覆蓋E(G);團覆蓋集是用團去覆蓋V(G)等等。一般說來,我們希望用最少個數的子圖去完成覆蓋,這就是最小覆蓋問題。另一類問題是研究從給定圖中,能取出多少個不交的某類子圖,或者反其意而說成是可以往給定圖中裝填多少個不交的某類子圖,這就是 裝填問題。一般說來,我們希望裝填的子圖個數儘可能地多,這就是 最大裝填問題。例如有向圖中弧形式的Menger定理所研究的是往D中裝填弧不交的()路。Menger定理是以最大最小定理的形式研究了能裝填k個弧不交的()路的充分必要條件 。
相關定理
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![裝填問題](/img/f/770/wZwpmL3IjNwcTO1IjN0YzM1UTM1QDN5MjM5ADMwAjMwUzLyYzL3UzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![裝填問題](/img/7/1db/wZwpmLzUzNwkzMwIDO0YzM1UTM1QDN5MjM5ADMwAjMwUzLygzLzczLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![裝填問題](/img/e/755/wZwpmL0UzMxUDO4kTN0YzM1UTM1QDN5MjM5ADMwAjMwUzL5UzL0UzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
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![裝填問題](/img/3/fbe/wZwpmL3MTNwITOwYjN0YzM1UTM1QDN5MjM5ADMwAjMwUzL2YzL3UzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
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給圖,對任意的,記{含過e的圈},定義稱為E-的閉集。一個子集,若不含圈,且,則稱為的一個基集。不難證明,的任何兩個基集,總包含相同個數的邊,定義為的基集所含的邊數。
![裝填問題](/img/b/ce7/wZwpmL3QjMwMzNzQTMxMzM1UTM1QDN5MjM5ADMwAjMwUzL0EzLxMzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![裝填問題](/img/6/b18/wZwpmL2YTM3kDO0IzN0YzM1UTM1QDN5MjM5ADMwAjMwUzLyczLyAzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
定理1(Tutte,Nash-Williams,1961)中含有k個邊不交的支撐樹的充分必要條件是對任何,
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定理1可以寫成另一種形式。
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![裝填問題](/img/7/99e/wZwpmL3IjN3ETO3MTN0YzM1UTM1QDN5MjM5ADMwAjMwUzLzUzL2IzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
定理1' 圖有k個邊不交的支撐樹,若且唯若對V的任意剖分,有
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需要指出的是定理1(1')對多重圖也是成立的,利用定理1,我們可以得到覆蓋邊集合所需要的森林的最小個數(記之為a(G),稱為圖G的蔭度) 。
![裝填問題](/img/c/48b/wZwpmL1UDN5czM5AjN0YzM1UTM1QDN5MjM5ADMwAjMwUzLwYzLxUzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
定理2 令G是非平凡的多重圖,是G中m階子圖的最大邊數,令
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則
![裝填問題](/img/2/b8e/wZwpmLxgDMyYzN4QzN0YzM1UTM1QDN5MjM5ADMwAjMwUzL0czL1AzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)