Minimum Non‐Chromatic‐λ $\lambda $‐Choosable Graphs

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Abstract
For a multiset of positive integers, let . A ‐list assignment of is a list assignment of such that the colour set can be partitioned into the disjoint union of sets so that for each and each vertex of , . We say is ‐choosable if is ‐colourable for any ‐list assignment of . The concept of ‐choosability puts ‐colourability and ‐choosability in a same framework: If , then ‐choosability is equivalent to ‐choosability; if consists of copies of 1, then ‐choosability is equivalent to ‐colourability. If is ‐choosable, then is ‐colourable. On the other hand, there are ‐colourable graphs that are not ‐choosable, provided that contains an integer larger than 1. Let be the minimum number of vertices in a ‐colourable non‐‐choosable graph. This paper determines the value of for all .

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