NoteIndependence and matching number in graphs with maximum degree 4

We prove that 74α(G)+β(G)≥n(G) and α(G)+32β(G)≥n(G) for every triangle-free graph G with maximum degree at most 4, where α(G) is the independence number and β(G) is the matching number of G, respectively. These results are sharp for a graph on 13 vertices. Furthermore we show χ(G)≤74ω(G) for {3K1,K1∪K5}-free graphs, where χ(G) is the chromatic number and ω(G) is the clique number of G, respectively.