Induced odd cycle packing number, independent sets, and chromatic number

2020/01/08 by Zdenĕk Dvořák, Dvořák, Zdeněk, Jakub Pekárek +1
Computer Science · Mathematics · #Advanced Graph Theory Research #Combinatorics (math.CO) #Data Structures and Algorithms (cs.DS) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Limits and Structures in Graph Theory #Markov Chains and Monte Carlo Methods

paper · pdf · doi:10.48550/arxiv.2001.02411

openalex publication_date 2020/01/08 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28

Abstract

The induced odd cycle packing number iocp(G) of a graph G is the maximum integer k such that G contains an induced subgraph consisting of k pairwise vertex-disjoint odd cycles. Motivated by applications to geometric graphs, Bonamy et al.~\citeindoc proved that graphs of bounded induced odd cycle packing number, bounded VC dimension, and linear independence number admit a randomized EPTAS for the independence number. We show that the assumption of bounded VC dimension is not necessary, exhibiting a randomized algorithm that for any integers k≥ 0 and t≥ 1 and any n-vertex graph G of induced odd cycle packing number at most k returns in time O_k,t(n^k+4) an independent set of G whose size is at least α(G)-n/t with high probability. In addition, we present χ-boundedness results for graphs with bounded odd cycle packing number, and use them to design a QPTAS for the independence number only assuming bounded induced odd cycle packing number.

Related

• The odd independence number of graphs, I: Foundations and classical classes
• The odd independence number of graphs, II: Finite and infinite grids and chessboard graphs
• Odd-Cycle-Packing-treewidth: On the Maximum Independent Set problem in odd-minor-free graph classes
• Tree decompositions with bounded independence number: beyond independent sets
• ALLSAT compressed with wildcards: All, or all maximum independent sets