English

Maximum $k$-colourable induced subgraphs in $(P_5+rK_1)$-free graphs

Combinatorics 2025-11-17 v2
Authors: Cicely Henderson , Evelyne Smith-Roberge , Sophie Spirkl , Rebecca Whitman
View on arXiv ↗ PDF ↗

Abstract

We show that for any nonnegative integer rr, the Weighted Maximum List-kk-Colourable Induced Subgraph problem can be solved in polynomial time for input graphs that do not contain (P5+rK1)(P_5+ rK_1) as an induced subgraph, and give an explicit algorithm demonstrating this. This answers a question of Agrawal et al.\ (2024).

Keywords

graph coloring

Cite

@article{arxiv.2410.08077,
  title  = {Maximum $k$-colourable induced subgraphs in $(P_5+rK_1)$-free graphs},
  author = {Cicely Henderson and Evelyne Smith-Roberge and Sophie Spirkl and Rebecca Whitman},
  journal= {arXiv preprint arXiv:2410.08077},
  year   = {2025}
}

Related papers

View all related →
Combinatorics · Mathematics
Maximum list $r$-colorable induced subgraphs in $kP_3$-free graphs
Esther Galby, Paloma T. Lima, Andrea Munaro, Amir Nikabadi
2025-05-05
Computational Complexity · Computer Science
Finding large $k$-colorable induced subgraphs in (bull, chair)-free and (bull,E)-free graphs
Nadzieja Hodur, Monika Pilśniak, Magdalena Prorok, Paweł Rzążewski
2025-04-15
Data Structures and Algorithms · Computer Science
Maximum Partial List H-Coloring on P_5-free graphs in polynomial time
Daniel Lokshtanov, Paweł Rzążewski, Saket Saurabh, Roohani Sharma +1
2026-03-06
Data Structures and Algorithms · Computer Science
Colouring $(P_r+P_s)$-Free Graphs
Tereza Klimošová, Josef Malík, Tomáš Masařík, Jana Novotná +2
2026-02-19
Combinatorics · Mathematics
Complexity dichotomy for List-5-Coloring with a forbidden induced subgraph
Sepehr Hajebi, Yanjia Li, Sophie Spirkl
2023-09-06
Combinatorics · Mathematics
List-$k$-Coloring $H$-free graphs for all $k>4$
Maria Chudnovsky, Sepehr Hajebi, Sophie Spirkl
2025-04-08
Data Structures and Algorithms · Computer Science
Deciding k-colourability of $P_5$-free graphs in polynomial time
Chính T. Hoàng, Marcin Kamiński, Vadim Lozin, J. Sawada +1
2016-08-14
Discrete Mathematics · Computer Science
k-Colorability of P5-free graphs
C. T. Hoang, J. Sawada, X. Shu
2007-05-23
Combinatorics · Mathematics
List-three-coloring graphs with no induced $P_6+rP_3$
Maria Chudnovsky, Shenwei Huang, Sophie Spirkl, Mingxian Zhong
2018-07-03
Discrete Mathematics · Computer Science
Polynomial-time algorithm for vertex k-colorability of P_5-free graphs
Marcin Kaminski, Vadim Lozin
2007-05-23
Data Structures and Algorithms · Computer Science
Induced subgraphs of bounded treewidth and the container method
Tara Abrishami, Maria Chudnovsky, Marcin Pilipczuk, Paweł Rzążewski +1
2020-03-12
Discrete Mathematics · Computer Science
Complexity of Coloring Graphs without Paths and Cycles
Pavol Hell, Shenwei Huang
2013-10-02
Combinatorics · Mathematics
4-coloring ($P_6$, bull)-free graphs
Frédéric Maffray, Lucas Pastor
2016-02-25
Discrete Mathematics · Computer Science
Polynomial-time algorithms for minimum weighted colorings of ($P_5, \bar{P}_5$)-free graphs and related graph classes
Chính T. Hoàng, D. Adam Lazzarato
2014-09-04
Data Structures and Algorithms · Computer Science
Sparse induced subgraphs in $P_7$-free graphs of bounded clique number
Maria Chudnovsky, Jadwiga Czyżewska, Kacper Kluk, Marcin Pilipczuk +1
2026-05-19
Combinatorics · Mathematics
Certifying coloring algorithms for graphs without long induced paths
Marcin Kamiński, Anna Pstrucha
2017-03-08
Discrete Mathematics · Computer Science
4-coloring $P_6$-free graphs with no induced 5-cycles
Maria Chudnovsky, Peter Maceli, Juraj Stacho, Mingxian Zhong
2014-07-10
Data Structures and Algorithms · Computer Science
Colouring $(sP_1+P_5)$-Free Graphs: a Mim-Width Perspective
Nick Brettell, Jake Horsfield, Daniel Paulusma
2020-04-27
Data Structures and Algorithms · Computer Science
Finding large $H$-colorable subgraphs in hereditary graph classes
Maria Chudnovsky, Jason King, Michał Pilipczuk, Paweł Rzążewski +1
2020-04-22
Combinatorics · Mathematics
3-Coloring $P_t$-Free Graphs With Only One Prescribed Induced Odd Cycle Length
Yidong Zhou, Mingxian Zhong, Shenwei Huang
2025-12-09
Data Structures and Algorithms · Computer Science
Polynomial-time algorithm for Maximum Weight Independent Set on $P_6$-free graphs
Andrzej Grzesik, Tereza Klimošová, Marcin Pilipczuk, Michał Pilipczuk
2020-03-24
Combinatorics · Mathematics
Coloring ($P_5$, bull)-free graphs
Frédéric Maffray
2017-07-28
Combinatorics · Mathematics
Approximately coloring graphs without long induced paths
Maria Chudnovsky, Oliver Schaudt, Sophie Spirkl, Maya Stein +1
2016-06-14
Data Structures and Algorithms · Computer Science
List $k$-Colouring $P_t$-Free Graphs: a Mim-width Perspective
Nick Brettell, Jake Horsfield, Andrea Munaro, Daniel Paulusma
2021-08-27
Combinatorics · Mathematics
Four-coloring $P_6$-free graphs. I. Extending an excellent precoloring
Maria Chudnovsky, Sophie Spirkl, Mingxian Zhong
2018-07-16
R2 v1 2026-06-28T19:16:33.153Z