ON LINEAR VERTEX-ARBORICITY OF COMPLEMENTARY GRAPHS

The linear vertex-arboricity rho'(G) of a graph G is defined to be the minimum number of subsets into which the vertex set of G can be partitioned such that each subset induces a linear forest. In this paper, we give the sharp upper and lower bounds for the sum and product of linear vertex-arboricities of a graph and its complement. Specifically, we prove that for any graph G of order p, rho'(G) + rho'(GBAR) less-than-or-equal-to 1 + [(p + 1)/2], rho'(G) . rho'(GBAR) less-than-or-equal-to [([(p + 3)/2]/2)2], and for any graph G of order p = (2n + 1)2, where n is-an-element-of Z+, 2n + 2 less-than-or-equal-to rho'(G) + rho'(GBAR). (C) 1994 John Wiley & Sons, Inc.