Induced Matchings in Graphs of Bounded Maximum Degree

Abstract

For a graph G, let ν s (G) be the induced matching number of G. We prove the sharp bound ν s (G) ≥ n(G) 9 for every graph G of maximum degree at most 4 and without isolated vertices that does not contain a certain blown up 5-cycle as a component. This result implies a consequence of the well known conjecture of Erdős and Nešetřil, saying that the strong chromatic index. Furthermore, it is shown that there is polynomial-time algorithm that computes induced matchings of size at least n(G) 9 .

A Stronger Bound for the Strong Chromatic Index

“…This is because induced matchings are much harder to handle than ordinary matchings. While the size of a largest matching can be determined quite precisely, it is hard even to obtain good bounds for induced matchings; see for instance [12,13,14].…”

A stronger bound for the strong chromatic index

“…This is because already induced matchings are much harder to handle than ordinary matchings. While the size of a largest matching can be quite precisely be determined, it is even hard to obtain good bounds for induced matchings; see for instance [11,12].…”

Linear programming based approximation for unweighted induced matchings --- breaking the $Δ$ barrier

“…If G arises from K ⌈ ∆ 2 ⌉+1 by attaching ⌊ ∆ 2 ⌋ many leaves to each vertex, then ν s (G) = 1 while any solution satisfying the above condition has total weight at least c ⌈ ∆ 2 ⌉ + 1 ⌊ ∆ 2 ⌋. This is also not surprising as combining the best-possible lower bound on the induced matching number for graphs of bounded maximum degree [7] with the best-possible Theorem 1 does not lead to an improved approximation ratio; both results are tight for different graphs.…”

“…Clearly, we may assume that G has no isolated vertices. We will describe an efficient recursive algorithm that constructs an induced matching M in G together with a feasible solution (y e ) e∈E(G) of the linear program (D) such that (i) y(δ G (u)) ≥ 1 3 for every vertex u of degree at most 2 in G, and (ii) y(E(G)) ≤ 7 3 |M |. We call the pair M, (y e ) e∈E(G) a good solution pair for G.…”

“…(e) y (E(H)) ≤ 7 3 . Condition (i) for the good solution pair P ′ for G ′ and condition (b) together imply y (δ G (e)) ≥ 1 for every edge e of G between V (H) and V (G ′ ).…”