Independence and matching number in graphs with maximum degree 4

We prove that 7/4 alpha(G) beta(G) >= n(G) and alpha(G) + 3/2 beta(G) >= n(G) for every triangle-free graph G with maximum degree at most 4, where alpha(G) is the independence number and beta(G) is the matching number of G, respectively. These results are sharp for a graph on 13 vertices. Furthermore we show chi(G) <= 7/4 omega(G) for {3K(1), K-1 boolean OR K-5}-free graphs, where chi(G) is the chromatic number and omega(G) is the clique number of G, respectively. (C) 2014 Elsevier B.V. All rights reserved.