Non-Path Spectrum Sets Guantao Chen, 1 Ralph. J. Faudree, 2 Xuechao Li, 3 and Ingo Schiermeyer 4 1 DEPARTMENT OF MATHEMATICS AND STATISTICS GEORGIA STATE UNIVERSITY ATLANTA, GEORGIA 30303 E-mail: gchen@mathstat.gsu.edu 2 THE UNIVERSITY OF MEMPHIS MEMPHIS, TENNESSEE 38152 E-mail: rfaudree@memphis.edu 3 ACADEMIC ENHANCEMENT THE UNIVERSITY OF GEORGIA ATHENS, GEORGIA 30602 E-mail: xcli@uga.edu 4 INST. FUR MATHEMATIK & INFORMATIK TU BERGAKADEMIE FRELBERG FREIBERG D-09596, GERMANY E-mail: schierme@math.tu-freiberg.de Received September 20, 2003; Revised February 22, 2008 Published online 23 May 2008 in Wiley InterScience(www.interscience.wiley.com). DOI 10.1002/jgt.20315 Abstract: A path of a graph is called maximal if it is not a proper subpath of any other path of the graph. The path spectrum of a graph G, denoted by ps(G), is the set of lengths of all maximal paths in the graph. A set S of positive integers is called a path spectrum if there is a connected graph G such that ps(G) = S. Jacobson et al. showed that all sets of positive integers with cardinality of 1 or 2 are path spectrum sets. Their results raised the question of whether all sets of positive integers are path spectra. We show that, for every positive integer k ≥ 3, there are infinitely many sets of k positive integers which are not path spectra. A set S of positive integers is called an absolute path spectrum if there are infinitely many connected Contract grant sponsor: NSF; Contract grant number: DMS-0070059. Journal of Graph Theory © 2008 Wiley Periodicals, Inc. 329