Complexity of computing the anti-Ramsey numbers
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The anti-Ramsey numbers are a fundamental notion in graph theory, introduced in 1978, by Erdös, Simonovits and Sós. For given graphs G and H the anti-Ramsey number ar(G,H) is defined to be the maximum number k such that there exists an assignment of k colors to the edges of G in which every copy of H in G has at least two edges with the same color. Precolored version of the problem is defined in a similar way except that the input graph is given with some fixed colors on some of the edges. Usually, combinatorists study extremal values of anti-Ramsey numbers for various classes of graphs. In this paper we study the complexity of computing the anti-Ramsey number ar(G,P_k), where P_k is a path of length k. First we observe the hardness of the problem when k is not fixed and we study the exact complexity of precolored version and show that there is no subexponential algorithm for the problem unless ETH fails already for k=3. We show that computing the ar(G,P_3) is hard to approximate to a factor of n^- 1/2 - ϵ even in 3-partite graphs, unless NP= ZPP. On the positive side we provide polynomial time algorithm for trees and we show approximability of the problem on special classes of graphs.
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