一個圖型(或超圖)的著色是適合對給定的邊著色，如果沒有邊和他的兩個端點著同色。ㄧ個圖的適合的著色數是指最小的使得對任意使用個顏色的邊著色都有一個使用相同個顏色的適合的點著色。 GGkkk 過去有關適合著色數的討論，零散的出現在各不同的問題上。有許多結果在各個不同的地方被證明。讓我們對這樣的問題感到興趣，而且這問題和原始的著色定義有很大的關連，所以可以將這問題和原始著色作一些比較。和原始著色的定義相比較，很容易看出對一個圖的適合的著色數會小於等於該圖原始的著色數。然而，階的乘積圖(categorical)的原始的著色數和的原始著色數是一樣的，而當我們看的適合著色數的時候是有可能隨著增加。在[12]我們猜想當充分大的時候)。當是完全圖的時候，我們證明這個猜想是對的當。如果是對特地的的時候的情況也是對的。但是還有很多問題是還沒被解決的在[12]，本計畫主要想解決[12]裡面提到的問題。 kkGGkGkk()(kadGGχχ=G4≥kn2=k 另外，適合著色數在其他類圖或和其他著色問題的比較，或增加一些特殊條件(如圍長或度數)，還有很多不清楚的地方。這些也是本計畫努力的方向。 A colouring of the vertices of a graph (or hypergraph) G is adapted to a given colouring of the edges of G if no edge has the same colour as both (or all) its vertices. The adaptable chromatic number of G is the smallest integer k such that each edge colouring of G by colours 1, 2, . . . , k admits an adapted vertex-colouring of G by the same colours 1, 2, . . . , k. (The adaptable chromatic number is just one more than a previously investigated notion of chromatic capacity.) The adaptable chromatic number of a graph G is smaller than or equal to the ordinary chromatic number of G. Consider the ordinary chromatic number of all (categorical) powers of G remains the same as that of G, the adaptable chromatic number of may increase with k. In [12], we conjecture that for all sufficiently large k the adaptable chromatic number of equals the chromatic number of G. When G is complete, we prove this conjecture with , and offer additional evidence suggesting it may hold with . In [12], we also discuss other products and propose several open problems. In this project, we focus on the problems proposed in [12]. kGkGkG4≥k2≥k Besides, there are still many problems related to adaptable chromatic number, that are not very clear. These are also the directions of this project.
