P3-factorization of triangulated Cartesian product of complete graphs

Abstract

Partition of G into edge-disjoint H-factors is called an H-factorization of G. Shehzad Afzal and Clemens Brand have factorized graphs into factors which are isomorphic to triangulated Cartesian product of two subgraphs. Also they have discussed properties of triangulated Cartesian product of more than two graphs. In this paper, we show that the necessary conditions mn ≡ 0 (mod 3), m, n are odd and 3(mn + m + n - 3) ≡ 0 (mod 8) are sufficient for the existence of a P3-factorization of Km ⧅ Kn, where ⧅ denotes triangulated Cartesian product of graphs, if one of the following holds: (i) m ≡ 9 (mod 12), n = 5, 13 and 17, (ii) m ≡ 9 (mod 12), ns, s > 1, n = 5, 13 and 17, (iii) m ≡ 9 (mod 12), n = psqt for all s, t ≥ 1, where p = 5, 13 and q = 13, 17, p ≠ q, (iv) m ≡ 9 (mod 12), n ≡ 9 (mod 12).