Decompositions of line graphs of complete graphs into paths and cycles

Let L(K-n) denote the line graph of the complete graph Kn. Let alpha and beta, beta &NOTEQUexpressionL; 1, be non -negative integers. Cox and Rodger (1996) [13] raised the following question: for what values of m and n does there exist an m-cycle decomposition of L(K-n)? In this paper, the above question is answered for m = p, where p is an odd prime. In fact, much more than this has been proved. Particularly, for any prime p >= 7, L(K-n) can be decomposed into alpha cycles of length p and beta paths of length p if and only if p(alpha + beta) = n( (2) (n-1 )),the number of edges of L(K-n). Consequently, when beta = 0, this completely characterizes the existence of a p-cycle decomposition of L(K-n), where p is an odd prime. Further, a complete characterization for the existence of a Pk+1-decomposition of L(K-n), where for k = p(l) , l >= 1 and p is an odd prime, is obtained. (c) 2022 Elsevier B.V. All rights reserved.