Large 2P3-free graphs with bounded degree

AbstractLet ex∗ (D;H) be the maximum number of edges in a connected graph with maximum degree D and no induced subgraph H; this is finite if and only if H is a disjoint union of paths. If the largest component of such an H has order m, then ex∗(D; H) = O(D2ex∗(D; Pm)). Constructively, ex∗(D;qPm) = Θ(gD2ex∗(D;Pm)) if q>1 and m > 2(Θ(gD2) if m = 2). For H = 2P3 (and D ⩾ 8), the maximum number of edges is 18[D4 + D3 + D2 + 6D] if D is even and 18 [D4 + D3 + 2D2 + 3D + 1 ] if D is odd, achieved by a unique extremal graph