Graphs with forbidden subgraphs and leaf number

Abstract

Let G be a simple connected, paw-free graph with minimum degree $$\delta $$ , leaf number L(G) and order n. We show for $$\delta \ge 3$$ that $$L(G)\ge \left\lceil \frac{1}{3}n+\frac{4}{3}\right\rceil $$ and the bound is sharp. Further, if G is claw-free, paw-free, we prove for $$\delta \ge 3$$ that $$L(G)\ge \left\lceil \frac{n}{2}+\frac{1}{2}\right\rceil $$ , and for $$\delta \ge 5$$ we prove that $$L(G)\ge \left\lceil \frac{4}{7}n+\frac{6}{7}\right\rceil $$ . The results apart from improving some of the results in Griggs et al. (J Graph Theory 13:669–695, 1988), Griggs and Wu (Discrete Math 104:167–183, 1992), Kleitman and West (SIAM J Discrete Math 4, 99–106, 1991), provide new lower bounds on leaf number whose practical applications in network designs are legion.