Repetition-freeness with Cyclic Relations and Chain Relations

Authors:
Tomi Kärki
Affiliations:
(Correspd.) Department of Teacher Education, University of Turku, PO Box 175, 26101 Rauma, Finland and Department of Mathematics, University of Turku, 20014 Turku, Finland, topeka@utu.fi
Venue:
Fundamenta Informaticae - Words, Graphs, Automata, and Languages; Special Issue Honoring the 60th Birthday of Professor Tero Harju
Year:
2012

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Abstract

A similarity relation R is a relation on words of equal length induced by a symmetric and reflexive relation on letters. Such a relation is called cyclic if the graph of the relation on letters is a cycle. A chain relation is obtained from a cyclic relation by removing one symmetric relation from the cycle. A word uv is an R-square if u and v are in relation R. The avoidability index of R-squares is the size of the minimal alphabet such that there exists an R-square-free infinite word having infinitely many occurrences of each letter of the alphabet. We prove that the avoidability index of R-squares is 7 in the case of cyclic relations and 6 in the case of chain relations. We also consider R-overlaps and show that they are 5-avoidable with cyclic relations and 4-avoidable with chain relations.