Factors and Factorization

Graphs and Combinatorics

Joos

Löwenstein

et al. 2016

Section: Notationmentioning

confidence: 99%

Induced Cycles in Graphs

Graphs and Combinatorics

Joos

Löwenstein

et al. 2016

“…If

M

is a matching of

G

, a vertex is

M

‐ matched if it is incident with an edge of

M

; otherwise, the vertex is

M

‐ unmatched . Matchings in graphs are extensively studied in the literature (see, eg, the classical book on matchings my Lovász and Plummer [9], and the excellent survey articles by Plummer [12] and Pulleyblank [13]).…”

Section: Introductionmentioning

confidence: 99%

A characterization of the subcubic graphs achieving equality in the Haxell‐Scott lower bound for the matching number

Journal of Graph Theory

Shozi

2020

In 2004, Biedl et al proved that if G is a connected cubic graph of order n, then α′(G)≥19(4n−1), where α′(G) is the matching number of G. The graphs achieving equality in this bound were characterized in 2010 by O and West. In 2017, Haxell and Scott proved that if G is a connected subcubic graph, then α′(G)≥49n3(G)+39n2(G)+29n1(G)−19, where ni(G) denotes the number of vertices of degree i in G. In this paper, we characterize the graphs achieving equality in the lower bound on the matching number given by Haxell and Scott.

“…Matchings in graphs are extensively studied in the literature (see, e.g. the classical book on matchings my Lovász and Plummer [11], and the excellent survey articles by Plummer [12] and Pulleyblank [13]). For ≥ 3, let be the set of all pairs ( , ) of real numbers for which there exists a constant such that…”

Section: Introductionmentioning

confidence: 99%

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

Journal of Graph Theory

Yeo

2018

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.