Matching and edge-connectivity in regular graphs

Self Cite

For positive integers, r ≥ 3, h ≥ 1, and k ≥ 1, Bollobás, Saito, and Wormald proved some sufficient conditions for an h‐edge‐connected r‐regular graph to have a k‐factor in 1985. Lu gave an upper bound for the third largest eigenvalue in a connected r‐regular graph to have a k‐factor in 2010. Gu found an upper bound for certain eigenvalues in an h‐edge‐connected r‐regular graph to have a k‐factor in 2014. For positive integers a ≤ b, an even (or odd) [ a , b ]‐factor of a graph G is a spanning subgraph H such that for each vertex v ∈ V ( G ), d H ( v ) is even (or odd) and a ≤ d H ( v ) ≤ b. In this paper, we prove upper bounds (in terms of a , b , and r) for certain eigenvalues (in terms of a , b , r , and h) in an h‐edge‐connected r‐regular graph G to guarantee the existence of an even [ a , b ]‐factor or an odd [ a , b ]‐factor. This result extends the one of Bollbás, Saito, and Wormald, the one of Lu, and the one of Gu.

“…In a subsequent paper [20], we use Theorem 2.1 to prove sharp upper bounds for certain eigenvalues in an

h

‐edge‐connected graph with given minimum degree to have a parity factor. With the method in the subsequent paper, we can show that the bounds in Theorem 1.3 are sharp by adjusting the parameter

t

of a graph in [33] (See Section 5) and the number of bullets

B_{r, t}

B_{r, t}^{'}

(See the definitions in [33]) depending on the parity of

r

and

t

.…”

Section: Discussionmentioning

confidence: 99%

Eigenvalues and [a,b]‐factors in regular graphs

Suil

2021

Self Cite

Covering 2‐connected 3‐regular graphs with disjoint paths

2017

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

Henning

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.