Independence and matching number in graphs with maximum degree 4

We prove that 7 4 α ( G ) + β ( G ) ≥ n ( G ) and α ( G ) + 3 2 β ( 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 ) ≤ 7 4 ω ( G ) for { 3 K 1 , K 1 ∪ K 5 } -free graphs, where χ ( G ) is the chromatic number and ω ( G ) is the clique number of G , respectively.