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Given a list L(v) for each vertex v, we say that the graph G is L-colorable if there is a proper vertex coloring of G where each vertex v takes its color from L(v). The graph is uniquely k-list colorable if there is a list assignment L such that |L(v)| = k for every vertex v and the graph has exactly one L-coloring with these lists. In this paper, we characterize uniquely list colorability of the graph G = K 2 + Om. We shall prove that if n = 2 then G is uniquely 3-list colorable if and only if m ≥ 9, if n = 3 and m ≥ 1 then G is uniquely 3-list colorable, if n ≥ 4 then G is uniquely k-list colorable with k = ⌊ m 2 ⌋ + 1 and if m ≥ n− 1 then G = K 2 +Om is UnLC.

Uniquely list colorability of the graph Kn-2+Om