Induced Path Factors Of Regular Graphs

An induced path factor of a graph G is a set of induced paths in G with the property that every vertex of G is in exactly one of the paths. The induced path number rho ( G ) of G is the minimum number of paths in an induced path factor of G. We show that if G is a connected cubic graph on n > 6 vertices, then rho ( G ) <= ( n - 1 ) / 3. Fix an integer k > 3. For each n, define M n to be the maximum value of rho ( G ) over all connected k-regular graphs G on n vertices. As n -> infinity with n k even, we show that c k = lim ( M n / n ) exists. We prove that 5 / 18 <= c 3 <= 1 / 3 and 3 / 7 <= c 4 <= 1 / 2 and that c k = 1 2 - O ( k - 1 ) for k -> infinity.