Work

There Are Finitely Many 5‐Vertex‐Critical (P6,bull)‐Free Graphs

Year: 2026

Type: article

Abstract: ABSTRACT In this paper, we are interested in 4‐colouring algorithms for graphs that do not contain an induced path on six vertices nor an induced bull, that is, the graph with vertex set and edge set . Such graphs are referred to as ‐free graphs. A graph is ‐ vertex‐critical if , and every proper induced subgraph of has . In the current paper, we investigate the structure of 5‐vertex‐critical ‐free graphs and show that there are only finitely many such graphs, thereby answering a question of Maffray and Pastor. A direct corollary of this is that there exists a polynomial‐time algorithm to decide if a ‐free graph is 4‐colourable such that this algorithm can also provide a certificate that can be verified in polynomial time and serves as a proof of 4‐colourability or non‐4‐colourability.

Source: Journal of Graph Theory

DOI: 10.1002/jgt.70027

Authors: Yiao Ju, Jorik Jooken, Jan Goedgebeur, Shenwei Huang

Institutions: Nankai University, KU Leuven, Ghent University

Language: English

Topic: Advanced Graph Theory Research

Subfield: Computational Theory and Mathematics

Field: Computer Science

Domain: Physical Sciences

Open Access status: green

Funders: National Natural Science Foundation of China, Fonds Wetenschappelijk Onderzoek, KU Leuven

Awards: G4256961292, G4261099548, Finding large independent sets in graphs with bounded degrees, G7294550521, Building bridges between computer-assisted graph theory and operations research