P5-FACTORIZATION OF CARTESIAN PRODUCT OF GRAPHS

Abstract

Partition of G into edge-disjoint H-factors is called H-factorization of G. Muthusamy and Paulraja have conjectured that for k ≥ 3, Km□Kn has a Pk-factorization if and only if mn ≡ 0 ( mod k) and k(m + n - 2) ≡ 0 ( mod 2(k - 1)), where □ denote cartesian product of graphs. In this paper, it is shown that the necessary conditions mn ≡ 0 ( mod 5) and 5(m + 1) ≡ 0 ( mod 8) are sufficient for the existence of a P5-factorization of Km□Cn when (i) m = 7, n ≡ 0 ( mod 5), (ii) m ≡ 175 ( mod 280), n ≡ 0( mod 5). Further, it is shown that the necessary conditions mn ≡ 0 ( mod 5) and 5(m + n - 2) ≡ 0 ( mod 8) are sufficient for the existence of a P5-factorization of Km□Kn in the following cases: (i) m ≡ 5 ( mod 40), n ≡ 5 ( mod 40), (ii) m ≡ 10 ( mod 40), n ≡ 0 ( mod 40), (iii) m ≡ 15 ( mod 120), n = 3 and m ≡ 15 ( mod 120), n ≡ 75 ( mod 120), (iv) m ≡ 20 ( mod 40), n ≡ 30 ( mod 120), (v) m ≡ 25 ( mod 40), n ≡ 25 ( mod 40), (vi) m ≡ 30 ( mod 120), n = 4 and m ≡ 30 ( mod 120), n ≡ 20 ( mod 40), (vii) m ≡ 75 ( mod 120), n ≡ 15 ( mod 120), (viii) m ≡ 0 ( mod 40), n = 2 and m ≡ 0 ( mod 40), n ≡ 10 ( mod 40). In fact our results partially answer the above conjecture when k = 5.