Large 2 P3-free graphs with bounded degree

Abstract

Let 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(D 2ex ∗(D; P m)) . Constructively, ex ∗(D;qP m) = Θ( gD 2ex ∗(D;P m)) if q>1 and m > 2( Θ( gD 2) if m = 2). For H = 2 P 3 (and D ⩾ 8), the maximum number of edges is 1 8 [D 4 + D 3 + D 2 + 6D] if D is even and 1 8 [D 4 + D 3 + 2D 2 + 3D + 1 ] if D is odd, achieved by a unique extremal graph.