A lower bound for the convexity number of some graphs

Abstract

Given a connected graph G, we say that a set C ∈ V(G) is convex in G if, for every pair of vertices x, y ∈ C, the vertex set of every x - y geodesic in G is contained in C. The convexity number of G is the cardinality of a maximal proper convex set in G. In this paper, we show that every pair k, n of integers with 2 ≤ k ≤ n - 1 is realizable as the convexity number and order, respectively, of some connected triangle-free graph, and give a lower bound for the convexity number of k-regular graphs of order n with n > k + 1.