Abstract
The antiRamsey numbers are a fundamental notion in graph theory, introduced in 1978, by Erdös, Simonovits and Sós. For given graphs G and H the antiRamsey 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.
Usually, combinatorists study extremal values of antiRamsey numbers for various classes of graphs. There are works on the computational complexity of the problem when H is a star. Along this line of research, we study the complexity of computing the antiRamsey number ar(G,P_k), where P_k is a path of length k. First, we observe that when k is close to n, the problem is hard; hence, the challenging part is the computational complexity of the problem when k is a fixed constant.
We provide a characterization of the problem for paths of constant length. Our first main contribution is to prove that computing ar(G,P_k) for every integer k > 2 is NPhard. We obtain this by providing several structural properties of such coloring in graphs. We investigate further and show that approximating ar(G,P₃) to a factor of n^{1/2  ε} is hard already in 3partite graphs, unless P = NP. We also study the exact complexity of the precolored version and show that there is no subexponential algorithm for the problem unless ETH fails for any fixed constant k.
Given the hardness of approximation and parametrization of the problem, it is natural to study the problem on restricted graph families. Along this line, we first introduce the notion of color connected coloring, and, employing this structural property, we obtain a linear time algorithm to compute ar(G,P_k), for every integer k, when the host graph, G, is a tree.
BibTeX  Entry
@InProceedings{akhoondianamiri_et_al:LIPIcs:2020:12678,
author = {Saeed Akhoondian Amiri and Alexandru Popa and Mohammad Roghani and Golnoosh Shahkarami and Reza Soltani and Hossein Vahidi},
title = {{Complexity of Computing the AntiRamsey Numbers for Paths}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {6:16:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771597},
ISSN = {18688969},
year = {2020},
volume = {170},
editor = {Javier Esparza and Daniel Kr{\'a}ľ},
publisher = {Schloss DagstuhlLeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2020/12678},
URN = {urn:nbn:de:0030drops126781},
doi = {10.4230/LIPIcs.MFCS.2020.6},
annote = {Keywords: Coloring, AntiRamsey, Approximation, NPhard, Algorithm, ETH}
}
Keywords: 

Coloring, AntiRamsey, Approximation, NPhard, Algorithm, ETH 
Collection: 

45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020) 
Issue Date: 

2020 
Date of publication: 

18.08.2020 