Metamorphosis of simple twofold triple systems into maximum twofold (K-4 - e)-packings

Let (X, B) be a simple twofold triple system of order v. For every x, y is an element of X, x not equal y, the pair {x, y} is contained in exactly two different triples, say, {x, y, z} and {x, y, w}. Any two blocks B_1, B_2 is an element of B satisfying vertical bar B_1 boolean AND B_2 vertical bar = 2 form a matched pair. Suppose that there is a partition of B into vertical bar B vertical bar/2 matched pairs. If we replace the double edge {x, y} with its corresponding single edge {x, y} from a matched pair {x, y, z}, {x, y, w}, we have a (K_4 - e) [x, y, z - w]. Let C be the collection of (K_4 - e)s obtained by replacing the double edge of each matched pair of B with its corresponding single edge, and F be the collection of the deleted edges. If F can be reassembled into a collection D of left perpendicularv(v - 1)/30right perpendicular (K_4 - e)s, then (X, C boolean OR D) is a maximum twofold (K_4 - e)-packing of order v. We call (X, C boolean OR D) a metamorphosis of the simple twofold triple system (X, B). In this paper, we show that there exists a metamorphosis of a simple twofold triple system of order v into a maximum twofold (K-4 - e)-packing of order v if and only if v equivalent to 0, 1 Let (X,B) be a simple twofold triple system of order v. For every x,y∈X, x≠y, the pair {x,y} is contained in exactly two different triples, say, {x,y,z} and {x,y,w}. Any two blocks B1,B2∈B satisfying |B1∩B2|=2 form a matched pair. Suppose that there is a partition of B into |B|/2 matched pairs. If we replace the double edge {x,y} with its corresponding single edge {x,y} from a matched pair {x,y,z}, {x,y,w}, we have a View the MathML source. Let C be the collection of (K4−e)s obtained by replacing the double edge of each matched pair of B with its corresponding single edge, and F be the collection of the deleted edges. If F can be reassembled into a collection D of ⌊v(v−1)/30⌋(K4−e)s, then (X,C∪D) is a maximum twofold (K4−e)-packing of order v. We call (X,C∪D) a metamorphosis of the simple twofold triple system (X,B). In this paper, we show that there exists a metamorphosis of a simple twofold triple system of order v into a maximum twofold (K4−e)-packing of order v if and only if View the MathML source and v≥4 with two exceptions of v=6,7 and one possible exception of v=18.