Unique list colorability of the graph Кⁿ₂ + K_r | Prikladnaya Diskretnaya Matematika - Applied Discrete Mathematics. 2022. № 55. DOI: 10.17223/20710410/55/6

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. If a graph G is not uniquely k-list colorable, we also say that G has property M(k). The least integer k such that G has the property M(k) is called the m-number of G, denoted by m(G). In this paper, we characterize the unique list colorability of the graph G = Kⁿ₂ + K_r. In particular, we determine the number m(G) of the graph G = Kⁿ₂ + K_r.