Uniquely list colorability of complete tripartite graphs
Xuan Hung Le Β· Π§Π΅Π±ΡΡΠ΅Π²ΡΠΊΠΈΠΉ ΡΠ±ΠΎΡΠ½ΠΈΠΊ Β· 2022
Given a list πΏ(π£) for each vertex π£, we say that the graph πΊ is πΏ-colorable if there is a proper vertex coloring of G where each vertex π£ takes its color from πΏ(π£). The graph is uniquely π-list colorable if there is a list assignment πΏ such that |πΏ(π£)| = π for every vertex π£ and the graph has exactly one πΏ-coloring with these lists. If a graph πΊ is not uniquely π-list colorable, we also say that πΊ has property π(π). The least integer π such that πΊ has the property π(π) is called the π-number of πΊ, denoted by π(πΊ). In this paper, first we characterize about the property of the complete tripartite graphs when it is uniquely π-list colorable graphs, finally we shall prove that π(πΎ2,2,π) = π(πΎ2,3,π) = π(πΎ2,4,π) = π(πΎ3,3,3) = 4 for every π > 9, π > 5, π > 4.