Tight lower bounds on the matching number in a graph with given maximum degree

Abstract

Let k≥3. We prove the following three bounds for the matching number, α′false(Gfalse), of a graph, G, of order n size m and maximum degree at most k. If k is odd, then α′false(Gfalse)≥false(k−1kfalse(k2−3false)false)n+false(k2−k−2kfalse(k2−3false)false)m−k−1k(k2−3). If k is even, then α′false(Gfalse)≥nk(k+1)+mk+1−1k. If k is even, then α′false(Gfalse)≥false(k+2k2+k+2false)m−false(k−2k2+k+2false)n−k+2k2+k+2. In this article, we actually prove a slight strengthening of the above for which the bounds are tight for essentially all densities of graphs. The above three bounds are in fact powerful enough to give a complete description of the set Lk of pairs (γ,β) of real numbers with the following property. There exists a constant K such that α′false(Gfalse)≥γn+βm−K for every connected graph G with maximum degree at most k, where n and m denote the number of vertices and the number of edges, respectively, in G. We show that Lk is a convex set. Further, if k is odd, then Lk is the intersection of two closed half‐spaces, and there is exactly one extreme point of Lk, while if k is even, then Lk is the intersection of three closed half‐spaces, and there are precisely two extreme points of Lk.

Transversals and Independence in Linear Hypergraphs with Maximum Degree Two

“…We shall need the following results by the authors [19] which establish a tight lower bound on the matching number of a graph in terms of its maximum degree, order, and size.…”

“…the electronic journal of combinatorics 24(2) (2017), #P2.50 Theorem 3. ( [19]) If k 3 is an odd integer and G is a connected graph of order n, size m, and with maximum degree ∆(G) k, then…”

“…Proposition 4. ( [19]) For k 3 an odd integer and r 1 arbitrary, if G ∈ F k,r has order n and size m, then…”

Lower Bounds on the Uniquely Restricted Matching Number

“…Again, (2) is tight for infinitely many trees [11] but we believe that it still holds if triangles are allowed and if ∆ ≥ 3. It is easy to see that every matching can be partitioned into at most ∆ acyclic matchings, which implies ν ur (G) ≥ ν ac (G) ≥ ν(G) ∆ .…”

“…If G is a tree, then (1) simplifies to ν ur (G) ≥ n−1 3 , which is tight for infinitely many trees [11]. Moreover, if the subcubic graph G k arises from the disjoint union of k isolated vertices u(1), .…”