Connected graphs without long paths

A problem, first considered by Erdős and Gallai [2], was to determine the maximum number of edges in any graph on n vertices if it contains no path with k + 1 vertices. This maximum number, ext(n, Pk+1), is called the extremal number for the path Pk+1. Erdős and Gallai proved the following theorem, which was one of the earliest extremal results in graph theory. Theorem 1.1 ([2]). For every k ≥ 0, ext(n, Pk+1) ≤ 12(k − 1)n with equality if and only if n = kt, in which case the extremal graph is ⋃t i=1 Kk. In 1975 this result was improved by Faudree and Schelp [3], determining ext(n, Pk+1) for all n > k > 0 as well as the corresponding extremal graphs. This is given by Theorem 1.2 ([3]). If G is a graph with |V (G)| = kt + r, 0 ≤ r < k, containing no path with k + 1 vertices then |E(G)| ≤ t(k 2 ) + ( r 2 ) with equality if and only if G is either (i) ( ⋃t i=1 Kk) ∪Kr, or (ii) ( ⋃t−l−1 i=1 Kk) ∪ (K(k−1)/2 + K(k+1)/2+lk+r) for some l, 0 ≤ l < t, when k is odd, t > 0, and r = (k ± 1)/2. We use G to denote the edge complement of a graph G, G ∪ H to denote the vertexdisjoint union of graphs G and H, and G + H to denote the join of G and H, defined as G ∪H together with all edges between G and H. In this paper we consider the extremal problem for Pk+1 taken over all connected graphs. We determine this number as well as the extremal graphs. These extremal graphs are particular examples of graphs of the following form. Definition. For n ≥ k > 2s > 0 let Gn,k,s = (Kk−2s ∪Kn−k+s) + Ks (see Figure 1).