On longest palindromic subwords of finite binary words

Abstract

In a recent paper, Müllner and Ryzhikov posed a question about palindromic subwords of finite binary words which can be rephrased as follows: Given four equally long binary words w1,w2,w3,w4 of total length n, what is the size of a longest palindrome p=qqR such that (i) q is a subword of w1w2 and also of (w3w4)R or (ii) q is a subword of w2w3 and also of (w4w1)R? Müllner and Ryzhikov conjectured that the answer is at least n/2. We disprove this conjecture, constructing sequences of words w1,w2,w3,w4 such that the longest palindromes have size 15n/32+o(n). Additionally, we show that the longest palindromes have size at least 3n/8.