G是一個圖，k是整數。對於所有的偶圈，我們刻劃出列表的充份條件使得偶圈可以(2k+1,k)-著色。其中條件1也是必要的條件，而條件2是精密的。

Suppose G is a graph and p >= 2q are positive integers. A
color-list is a mapping L: V --> P(0, 1,...,p-1) which assigns to each vertex a set L(v) of
permissible colors. An L-(p, q)-coloring of G is a (p,
q)-coloring h of G such that for each vertex v,
h(v) in L(v). We say G is L-(p, q)-colorable if
such a coloring exists. A color-size-list is a mapping f: V -->{0, 1, 2,..., p}, which assigns to each vertex v a
non-negative integer f(v). We say G is f-(p, q)-colorable
if for every color-list L with |{L}(v)| = f(v), G is
L-(p, q)-colorable. For odd cycles C, Raspaud and Zhu
gave a sharp sufficient condition for a color-size-list f under
which C is f-(2k+1, k)-colorable. The corresponding
question for even cycles remained open. In this paper, we
consider list circular coloring of even cycles. For each even cycle C of length n and for each positive integer k, we
give a condition on f which is sufficient and sharp for C to
be f-(2k+1, k)-colorable.

1. Introduction 1
2. Some notation 2
3. The main result and sharpness of the conditons 3
4. Some preliminaries 6
5. The structure of minimal counterexample 8
6. Proof of main theorem 16

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