Anti-Ramsey numbers for trees in complete multi-partite graphs

Abstract

Let G be a complete multi-partite graph of order n. In this paper, we consider the anti-Ramsey number ar(G,Tq) with respect to G and the set Tq of trees with q edges, where 2≤q≤n−1. For the case q=n−1, the result has been obtained by Lu, Meier and Wang. We will extend it to q<n−1. We first show that ar(G,Tq)=ℓq(G)+1, where ℓq(G) is the maximum size of a disconnected spanning subgraph H of G with the property that any two components of H together have at most q vertices. Using this equality, we obtain the exact values of ar(G,Tq) for n−3≤q≤n−1. Moreover, for the general case when (4n−2)/5≤q≤n−1, ar(G,Tq) can be determined by a simple algorithm. In particular, the explicit expression of ar(G,Tq) is given when G has a partite set much larger than all the other partite sets.