Randomly finding independent sets in locally sparse graphs

Ye Wang and Yan Li ()
Additional contact information
Ye Wang: Harbin Engineering University
Yan Li: University of Shanghai for Science and Technology

Journal of Combinatorial Optimization, 2023, vol. 45, issue 1, No 22, 7 pages

Abstract: Abstract For graph G with vertex set V and $$u\in V$$ u ∈ V , we write $$d_u$$ d u to be the degree of u. For $$\emptyset \ne S\subseteq V$$ ∅ ≠ S ⊆ V , the independence ratio of S is defined as $$\rho (S)=\min \{\frac{\alpha (T)}{|T|}:\;\emptyset \ne T\subseteq S\}$$ ρ ( S ) = min { α ( T ) | T | : ∅ ≠ T ⊆ S } , where $$\alpha (T)$$ α ( T ) is the independence number of the graph induced by T in G. Let $$N_i(u)=\{w\in V:\;dist(u,w)=i\}$$ N i ( u ) = { w ∈ V : d i s t ( u , w ) = i } . We shall show that if $$\rho _0>0$$ ρ 0 > 0 and a graph G has $$\rho (N_i(u))\ge \rho _0$$ ρ ( N i ( u ) ) ≥ ρ 0 for each $$u\in V$$ u ∈ V and $$i\le k$$ i ≤ k , then its independence number is at least $$a\left( \sum _{u}d_u^{1/(k-1)}\right) ^{(k-1)/k}$$ a ∑ u d u 1 / ( k - 1 ) ( k - 1 ) / k , where $$a=a(k,\rho _0)>0$$ a = a ( k , ρ 0 ) > 0 is independent of G. This result generalizes some former results for graphs with forbidden odd cycles.

Keywords: Independence number; (k; m)-colorable; Local sparseness (search for similar items in EconPapers)
Date: 2023
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s10878-022-00948-w Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:jcomop:v:45:y:2023:i:1:d:10.1007_s10878-022-00948-w

Ordering information: This journal article can be ordered from
https://www.springer.com/journal/10878

DOI: 10.1007/s10878-022-00948-w

Access Statistics for this article

Journal of Combinatorial Optimization is currently edited by Thai, My T.

More articles in Journal of Combinatorial Optimization from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().