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For a graph G, let vs(G) be the size of a largest induced matching of G. We prove that vs(G) ≥ n(G)/(⌈Δ/2⌇+1)(⌈Δ/2⌇+1) for every graph of sufficiently large maximum degree Δ and without isolated vertices. This bound is sharp. Moreover, there is a polynomial-time algorithm which computes induced matchings of the size stated above. [ABSTRACT FROM AUTHOR]

Extending results of Kobler and Rotics (2003), Cameron and Walker (2005) gave a complete structural description of the graphs G where the matching number ν ( G ) equals the induced matching number ν 2 ( G ) . We present a short proof of their result and use it to study graphs G with ν ( G ) − ν 2 ( G ) ≤ k . We show that the recognition of these graphs can be done in polynomial time for fixed k , and is fixed parameter tractable when parameterized by k for graphs of bounded maximum degree. [ABSTRACT FROM AUTHOR]

Abstract: We prove that and for every triangle-free graph with maximum degree at most , where is the independence number and is the matching number of , respectively. These results are sharp for a graph on vertices. Furthermore we show for -free graphs, where is the chromatic number and is the clique number of , respectively. [Copyright &y& Elsevier]

We prove that a cubic graph with m edges has an induced matching with at least M/9 edges. Our result generalizes a result for planar graphs due to Kang, Mnich, and Müller (SIAM J. Discrete Math., 26 (2012), pp. 1383-1411) and solves a conjecture of Henning and Rautenbach. [ABSTRACT FROM AUTHOR]